问题

### Instructions: You are an expert Python programmer. You will be given a question (problem specification) and will generate a correct Python program that matches the specification and passes all tests. You will NOT return anything except for the program.
### Question:
For a positive integer k, the Pyramid Sequence of size k is a sequence of length (2k-1) where the terms of the sequence have the values 1,2,\ldots,k-1,k,k-1,\ldots,2,1 in this order.
You are given a sequence A=(A_1,A_2,\ldots,A_N) of length N.
Find the maximum size of a Pyramid Sequence that can be obtained by repeatedly choosing and performing one of the following operations on A (possibly zero times).

- Choose one term of the sequence and decrease its value by 1.
- Remove the first or last term.

It can be proved that the constraints of the problem guarantee that at least one Pyramid Sequence can be obtained by repeating the operations.

Input

The input is given from Standard Input in the following format:
N
A_1 A_2 \ldots A_N

Output

Print the maximum size of the Pyramid Sequence that can be obtained by repeatedly performing the operations described in the problem statement on the sequence A.

Constraints


- 1\leq N\leq 2\times 10^5
- 1\leq A_i\leq 10^9
- All input values are integers.

Sample Input 1

5
2 2 3 1 1

Sample Output 1

2

Starting with A=(2,2,3,1,1), you can create a Pyramid Sequence of size 2 as follows:

- Choose the third term and decrease it by 1. The sequence becomes A=(2,2,2,1,1).
- Remove the first term. The sequence becomes A=(2,2,1,1).
- Remove the last term. The sequence becomes A=(2,2,1).
- Choose the first term and decrease it by 1. The sequence becomes A=(1,2,1).

(1,2,1) is a Pyramid Sequence of size 2.
On the other hand, there is no way to perform the operations to create a Pyramid Sequence of size 3 or larger, so you should print 2.

Sample Input 2

5
1 2 3 4 5

Sample Output 2

3

Sample Input 3

1
1000000000

Sample Output 3

1

### Format: Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within delimiters as follows.
```python
# YOUR CODE HERE
```

### Answer: (use the provided format with backticks)

答案

模型 模型答案 正确答案
金声玉亮2.0 ```python
import sys
input = sys.stdin.read
data = input().split()

N = int(data[0])
A = list(map(int, data[1:]))

def max_pyramid_size(A):
left = [0] * (N + 1)
right = [0] * (N + 1)

for i in range(N):
left[i + 1] = max(left[i], A[i])

for i in range(N - 1, -1, -1):
right[i] = max(right[i + 1], A[i])

max_size = 0
for i in range(N):
if left[i] <= right[i + 1]:
max_size = max(max_size, min(left[i], right[i + 1]))

return max_size

print(max_pyramid_size(A))
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
deepseek-r1

```python
n = int(input())
A = list(map(int, input().split()))

if n == 0:
print(0)
exit()

left = [0] * n
left[0] = min(1, A[0])
for i in range(1, n):
left[i] = min(left[i-1] + 1, A[i])

right = [0] * n
right[-1] = min(1, A[-1])
for i in range(n-2, -1, -1):
right[i] = min(right[i+1] + 1, A[i])

max_k = 0
for m in range(n):
if m + 1 < n:
current_right = right[m+1]
else:
current_right = 0
current_k = min(left[m], current_right + 1)
if current_k > max_k:
max_k = current_k

print(max_k)
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
gpto1 ```python
# YOUR CODE HERE
import sys
import threading

def main():
import sys
import math
import bisect
sys.setrecursionlimit(1 << 25)
N_and_rest = sys.stdin.read().split()
N = int(N_and_rest[0])
A = list(map(int, N_and_rest[1:N+1]))
N = len(A)
answer = 1 # At least k=1 can be achieved

left = 1
right = (N + 1) // 2

while left <= right:
mid = (left + right) // 2
k = mid
L = 2 * k - 1
if L > N:
right = mid - 1
continue

S = []
for i in range(k - 1):
S.append(i + 1)
S.append(k)
for i in range(k - 1):
S.append(k - i - 1)
# Pad S to length N with zeros
S += [0] * (N - L)
D = [A[i] - S[i] for i in range(N)]
# Sliding window minimum of size L
from collections import deque
min_queue = deque()
found = False
for i in range(N):
while min_queue and min_queue[-1] > D[i]:
min_queue.pop()
min_queue.append(D[i])
if i >= L:
if min_queue[0] == D[i - L]:
min_queue.popleft()
if i >= L - 1:
if min_queue[0] >= 0:
found = True
break
if found:
answer = k
left = mid + 1
else:
right = mid - 1
print(answer)
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
gpt-4o-2024-08-06 ```python
# YOUR CODE HERE
def max_pyramid_sequence_size(N, A):
# The maximum size of the pyramid sequence we can form is determined by the
# maximum k such that we can form a sequence 1, 2, ..., k, ..., 2, 1.
# This means we need to find the maximum k such that:
# 1 <= 1, 2 <= 2, ..., k <= k, ..., 2 <= 2, 1 <= 1
# This is equivalent to finding the maximum k such that:
# A[i] >= i for i in 1 to k and A[N-i+1] >= i for i in 1 to k

# We will use a binary search to find the maximum k.
def can_form_pyramid(k):
# Check if we can form a pyramid of size k
for i in range(1, k + 1):
if A[i - 1] < i or A[N - i] < i:
return False
return True

# Binary search for the maximum k
left, right = 1, (N + 1) // 2
while left < right:
mid = (left + right + 1) // 2
if can_form_pyramid(mid):
left = mid
else:
right = mid - 1

return left

import sys
input = sys.stdin.read
data = input().split()
N = int(data[0])
A = list(map(int, data[1:]))

print(max_pyramid_sequence_size(N, A))
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
DeepSeek-Coder-V2-Lite-Instruct ```python
# YOUR CODE HERE
def find_max_pyramid_sequence(N, A):
max_size = 0
for start in range(N):
for end in range(start, N):
B = A[:]
while start < end:
if B[start] > B[start + 1]:
B[start] -= 1
if B[end] > B[end - 1]:
B[end] -= 1
start += 1
end -= 1
max_size = max(max_size, end - start + 1)
return max_size

# Read input
import sys
input = sys.stdin.read
data = input().split()
N = int(data[0])
A = list(map(int, data[1:]))

# Solve the problem
max_size = find_max_pyramid_sequence(N, A)

# Output the result
print(max_size)
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
Meta-Llama-3.1-405B-Instruct-Turbo ```python
import sys

def solve():
n = int(input())
a = list(map(int, input().split()))

def check(k):
b = a[:]
i, j = 0, n - 1
for x in range(1, k):
if i >= j:
return False
if b[i] < x:
return False
if b[j] < x:
return False
b[i] -= x
b[j] -= x
i += 1
j -= 1
return True

l, r = 1, n + 1
while l < r:
mid = (l + r + 1) // 2
if check(mid):
l = mid
else:
r = mid - 1
print(l)

if __name__ == "__main__":
solve()
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
Meta-Llama-3.1-70B-Instruct-Turbo ```python
import sys

def solve():
n = int(input())
a = list(map(int, input().split()))

def check(k):
i, j = 0, n - 1
while i < j:
if a[i] < k - i or a[j] < k - i:
return False
if a[i] == k - i:
i += 1
elif a[j] == k - i:
j -= 1
else:
return False
return True

lo, hi = 1, n + 1
while lo < hi:
mid = (lo + hi) // 2
if check(mid):
lo = mid + 1
else:
hi = mid
print(lo - 1)

if __name__ == "__main__":
solve()
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
Qwen2-72B-Instruct ```python
from itertools import accumulate

N = int(input())
A = list(map(int, input().split()))

def pyramid_size(A):
# Calculate the maximum pyramid size from the left and right
left = [0] * N
right = [0] * N
stack = []
for i, a in enumerate(A):
while stack and A[stack[-1]] >= a:
stack.pop()
left[i] = stack[-1] if stack else -1
stack.append(i)
stack = []
for i in range(N-1, -1, -1):
while stack and A[stack[-1]] >= A[i]:
stack.pop()
right[i] = stack[-1] if stack else N
stack.append(i)

# Calculate the maximum pyramid size for each element
pyramid = [min(i - left[i], right[i] - i) for i in range(N)]
for i in range(1, N):
pyramid[i] = max(pyramid[i], pyramid[i-1] - 1)
for i in range(N-2, -1, -1):
pyramid[i] = max(pyramid[i], pyramid[i+1] - 1)

return pyramid

pyramid_left = pyramid_size(A)
pyramid_right = pyramid_size(A[::-1])[::-1]

# Calculate the maximum pyramid size by combining the left and right pyramid sizes
max_pyramid_size = max(pyramid_left[i] + pyramid_right[i] - 1 for i in range(N))

print(max_pyramid_size)
```

```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
baichuan4 ```python
# YOUR CODE HERE
import sys

def max_pyramid_sequence_size(n, sequence):
max_size = 0
for i in range(n):
# Calculate the maximum size of the Pyramid Sequence starting from index i
size = 1
left = sequence[i]
right = sequence[i]
j = i + 1
k = i - 1
while j < n and left >= sequence[j]:
left -= 1
size += 1
j += 1
while k >= 0 and right >= sequence[k]:
right -= 1
size += 1
k -= 1
max_size = max(max_size, size)
return max_size

# Read input from stdin
input_data = sys.stdin.read().splitlines()
n = int(input_data[0])
sequence = list(map(int, input_data[1].split()))

# Calculate the maximum size of the Pyramid Sequence
result = max_pyramid_sequence_size(n, sequence)

# Write the result to stdout
print(result)
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
doubao ```python
import bisect

def find_max_pyramid_size(a):
n = len(a)
dp = [0] * (n + 1)
for i in range(n):
for j in range(i + 1):
k = i - j
if a[j] > k:
continue
dp[i + 1] = max(dp[i + 1], dp[j] + min(a[j], k - a[j]) + 1)
return dp[n]

n = int(input())
a = list(map(int, input().split()))
print(find_max_pyramid_size(a))
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
gpto1 ```python
# YOUR CODE HERE
import sys
import threading

def main():
import sys
import math
import bisect
sys.setrecursionlimit(1 << 25)
N_and_rest = sys.stdin.read().split()
N = int(N_and_rest[0])
A = list(map(int, N_and_rest[1:N+1]))
N = len(A)
answer = 1 # At least k=1 can be achieved

left = 1
right = (N + 1) // 2

while left <= right:
mid = (left + right) // 2
k = mid
L = 2 * k - 1
if L > N:
right = mid - 1
continue

S = []
for i in range(k - 1):
S.append(i + 1)
S.append(k)
for i in range(k - 1):
S.append(k - i - 1)
# Pad S to length N with zeros
S += [0] * (N - L)
D = [A[i] - S[i] for i in range(N)]
# Sliding window minimum of size L
from collections import deque
min_queue = deque()
found = False
for i in range(N):
while min_queue and min_queue[-1] > D[i]:
min_queue.pop()
min_queue.append(D[i])
if i >= L:
if min_queue[0] == D[i - L]:
min_queue.popleft()
if i >= L - 1:
if min_queue[0] >= 0:
found = True
break
if found:
answer = k
left = mid + 1
else:
right = mid - 1
print(answer)
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
kimi ```python
# YOUR CODE HERE
def check(k, A):
l = 0
r = len(A) - 1
cnt = 0
while l < r:
if cnt < k:
if A[l] - A[r] >= k - cnt:
A[l] -= k - cnt
elif A[r] - A[l] >= k - cnt:
A[r] -= k - cnt
else:
return False
cnt += 1
l += 1
r -= 1
if cnt < k and A[l] >= k - cnt:
return True
return cnt == k

def binary_search(A):
left, right = 1, len(A) // 2 + 1
while left < right:
mid = (left + right) // 2
if check(mid, A[:]):
left = mid + 1
else:
right = mid
return left - 1

def main():
N = int(input())
A = list(map(int, input().split()))
print(binary_search(A))

def __starting_point():
main()
__starting_point()
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
minimax ```python
# YOUR CODE HERE
from collections import deque

def max_pyramid_size(N, A):
# Initialize a deque for efficient popping from both ends
sequence = deque(A)
# Counters for the pyramid size and the current level
pyramid_size, level = 0, 0
# The maximum value in the sequence
max_value = max(A)

while sequence:
# The number of elements needed for the current level
level_elements = 2 * level + 1
# Check if we have enough elements to form the current level
if len(sequence) < level_elements:
break

# Check if the current level can be formed with the available elements
if all(x >= level for x in sequence):
# Increase the pyramid size and move to the next level
pyramid_size += 1
level += 1
else:
# If the current level cannot be formed, remove the smallest element
sequence.remove(min(sequence))

return pyramid_size

# Read the inputs from stdin
N = int(input())
A = list(map(int, input().split()))

# Solve the problem and write the answer to stdout
print(max_pyramid_size		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
wenxinyiyan ```python
# Read the input
N = int(input())
A = list(map(int, input().split()))

# Initialize variables
max_size = 0
left = 0
right = 0
current_max = 0

# Loop through the array to find the maximum pyramid size
while right < N:
# Increase the right pointer and update the current maximum value
current_max = max(current_max, A[right])

# If the current maximum value is greater than or equal to the difference between the right and left pointers plus 1,
# we can potentially form a pyramid sequence
while current_max >= right - left + 1:
# Update the maximum pyramid size
max_size = max(max_size, 2 * (right - left) + 1)

# Move the left pointer and update the current maximum value
current_max = max(A[left+1:right+1])
left += 1

# Move the right pointer
right += 1

# Print the maximum pyramid size
print(max_size)
```
**Note**: The provided solution has a time complexity of O(N), where N is the length of the input sequence. It uses a sliding window approach to find the maximum pyramid size by maintaining two pointers (`left` and `right`) and a current maximum value (`current_max`) within the window. The solution repeatedly moves the right pointer, updates the current maximum value, and then moves the left pointer as long as a potential pyramid sequence can be formed. Finally, it prints the maximum pyramid size found during the process.		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
zhipu ```python
def main():
import sys
input = sys.stdin.read
data = input().split()

N = int(data[0])
A = list(map(int, data[1:]))

# This will hold the maximum size of the pyramid sequence we can find
max_pyramid_size = 0

# We use two pointers to try to form the largest possible pyramid sequence
left = 0
right = N - 1

# We will attempt to build the pyramid from the outside in
while left <= right:
# Current potential height of the pyramid
current_height = min(A[left], A[right])

# Check if we can form a valid pyramid with the current height
if left == right or A[left] == A[right]:
# Update the maximum size found so far
max_pyramid_size = max(max_pyramid_size, current_height)

# Move both pointers towards the center
left += 1
right -= 1
else:
# If the values at the pointers do not match, we need to adjust them
if A[left] < A[right]:
right -= 1
else:
left += 1

print(max_pyramid_size)

if __name__ == "__main__":
main()
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]
deepseek-v3 ```python
# YOUR CODE HERE
def max_pyramid_size(N, A):
left = 1
right = N
result = 1

while left <= right:
mid = (left + right) // 2
possible = True
current = 1
for i in range(N):
if A[i] >= current:
current += 1
else:
current = A[i] + 1
if current > mid:
break
if current > mid:
possible = False
else:
current = mid
for i in range(N-1, -1, -1):
if A[i] >= current:
current -= 1
else:
current = A[i] - 1
if current < 1:
break
if current >= 1:
possible = True
else:
possible = False

if possible:
result = mid
left = mid + 1
else:
right = mid - 1

return result

# Read input
N = int(input())
A = list(map(int, input().split()))

# Compute and print the result
print(max_pyramid_size(N, A))
```		 [{"input": "5\n2 2 3 1 1\n", "output": "2\n", "testtype": "stdin"}, {"input": "5\n1 2 3 4 5\n", "output": "3\n", "testtype": "stdin"}, {"input": "1\n1000000000\n", "output": "1\n", "testtype": "stdin"}]