#P1805F2. Survival of the Weakest (hard version)
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ID: 135
Type: RemoteJudge
3000ms
256MiB
Tried: 0
Accepted: 0
Difficulty: 10
Uploaded By:
Hydro
Tags> Survival of the Weakest (hard version)
Description
This is the hard version of the problem. It differs from the easy one only in constraints on $n$. You can make hacks only if you lock both versions.
Let $a_1, a_2, \ldots, a_n$ be an array of non-negative integers. Let $F(a_1, a_2, \ldots, a_n)$ be the sorted in the non-decreasing order array of $n - 1$ smallest numbers of the form $a_i + a_j$, where $1 \le i < j \le n$. In other words, $F(a_1, a_2, \ldots, a_n)$ is the sorted in the non-decreasing order array of $n - 1$ smallest sums of all possible pairs of elements of the array $a_1, a_2, \ldots, a_n$. For example, $F(1, 2, 5, 7) = [1 + 2, 1 + 5, 2 + 5] = [3, 6, 7]$.
You are given an array of non-negative integers $a_1, a_2, \ldots, a_n$. Determine the single element of the array $\underbrace{F(F(F\ldots F}_{n-1}(a_1, a_2, \ldots, a_n)\ldots))$. Since the answer can be quite large, output it modulo $10^9+7$.
The first line contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the initial length of the array.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array elements.
Output a single number — the answer modulo $10^9 + 7$.
Input
The first line contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the initial length of the array.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array elements.
Output
Output a single number — the answer modulo $10^9 + 7$.
5
1 2 4 5 6
9
1 1 1 7 7 7 9 9 9
7
1 7 9 2 0 0 9
3
1000000000 1000000000 777
34
256
20
1540
Note
In the first test, the array is transformed as follows: $[1, 2, 4, 5, 6] \to [3, 5, 6, 6] \to [8, 9, 9] \to [17, 17] \to [34]$. The only element of the final array is $34$.
In the second test, $F(a_1, a_2, \ldots, a_n)$ is $[2, 2, 2, 8, 8, 8, 8, 8]$. This array is made up of $3$ numbers of the form $1 + 1$ and $5$ numbers of the form $1 + 7$.
In the fourth test, the array is transformed as follows: $[10^9, 10^9, 777] \to [10^9+777, 10^9+777] \to [2 \cdot 10^9 + 1554]$. $2 \cdot 10^9 + 1554$ modulo $10^9+7$ equals $1540$.