Problem:
Score : 600 points

Problem Statement

Given is an integer sequence A_1, ..., A_N of length N.

We will choose exactly \left\lfloor \frac{N}{2} \right\rfloor elements from this sequence so that no two adjacent elements are chosen.

Find the maximum possible sum of the chosen elements.

Here \lfloor x \rfloor denotes the greatest integer not greater than x.

Constraints

2 \leq N \leq 2\times 10^5

|A_i|\leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N

Output

Print the maximum possible sum of the chosen elements.

Sample Input 1

6
1 2 3 4 5 6

Sample Output 1

12

Choosing 2, 4, and 6 makes the sum 12, which is the maximum possible value.

Sample Input 2

5
-1000 -100 -10 0 10

Sample Output 2

0

Choosing -10 and 10 makes the sum 0, which is the maximum possible value.

Sample Input 3

10
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000

Sample Output 3

5000000000

Watch out for overflow.

Sample Input 4

27
18 -28 18 28 -45 90 -45 23 -53 60 28 -74 -71 35 -26 -62 49 -77 57 24 -70 -93 69 -99 59 57 -49

Sample Output 4

295

Solution in PyPy3 (7.3.0) (judge verdict: Time Limit Exceeded)


import math

N = int(input())
num = math.floor(N/2)

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

# iまででj個を選び、状態がk（i番目を選んでいる（もしくはいない））の場合の最大値
# j : [(i-1)/2] -1, 0, 1

dp = [[[- float('inf')] * 2 for _ in range(N+1)] for _ in range(N+1)]
dp[0][0][0] = 0

for i in range(1, N+1):
    min_j = math.floor((i-1)/2) - 1
    # print(i, min_j)

    for j in range(3):
        # k = 1なら次は必ず選ばない（その場合jは変わらない）
        dp[i][min_j + j][0] = max(dp[i][min_j + j][0], dp[i-1][min_j + j][1])
        # k = 0なら選んでも選ばなくてもいい
        dp[i][min_j + j][0] = max(dp[i][min_j + j][0], dp[i-1][min_j + j][0])
        dp[i][min_j + j + 1][1] = max(dp[i][min_j + j + 1][1], dp[i-1][min_j + j][0] + A[i-1])


'''
# print(dp)
for i in range(1, N+1):
    min_j = math.floor((i-1)/2)
    print(i, min_j, dp[i][min_j-1:min_j + 2])
'''

print(max(dp[N][num][0], dp[N][num][1]))