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 Ruby (2.7.1) (judge verdict: Accepted)

require 'set'; require 'prime'; require 'pp'
INF=Float::INFINITY; MOD=10**9+7
n = gets.chomp.to_i
@a = gets.chomp.split.map(&:to_i)
@n = n
k = n/2

if n == 2 || n == 3
  puts @a.max
  exit
end

@memo = Array.new(3){Array.new(n+5)}
def solve(index,skip_c)
  @memo[skip_c][index] ||=
  if index >= @n
    if skip_c > 0
      -INF
    else
      0
    end
  elsif skip_c == 0
    solve(index + 2, 0) + @a[index]
  elsif skip_c == 1
    tmp = []
    [2, @n - index - 1].min.times do |i|
      tmp.push solve(index + 2 + i, skip_c - i) + @a[index + i]
    end
    # p [index, skip_c, tmp]
    tmp.max || -INF
  else
    tmp = []
    [3, @n - index - 1].min.times do |i|
      tmp.push solve(index + 2 + i, skip_c - i) + @a[index + i]
    end
    # p [index, skip_c, tmp]
    tmp.max || -INF
  end
end

if n.even?
  puts solve(0,1)
else
  puts solve(0,2)
end