1406D · Three Sequences

Problem: You are given a sequence of $n$ integers $a_1, a_2, \ldots, a_n$.

You have to construct two sequences of integers $b$ and $c$ with length $n$ that satisfy:

• for every $i$ ($1\leq i\leq n$) $b_i+c_i=a_i$
• $b$ is non-decreasing, which means that for every $1<i\leq n$, $b_i\geq b_{i-1}$ must hold
• $c$ is non-increasing, which means that for every $1<i\leq n$, $c_i\leq c_{i-1}$ must hold

You have to minimize $\max(b_i,c_i)$. In other words, you have to minimize the maximum number in sequences $b$ and $c$.

Also there will be $q$ changes, the $i$-th change is described by three integers $l,r,x$. You should add $x$ to $a_l,a_{l+1}, \ldots, a_r$.

You have to find the minimum possible value of $\max(b_i,c_i)$ for the initial sequence and for sequence after each change.

Input Format: The first line contains an integer $n$ ($1\leq n\leq 10^5$).

The secound line contains $n$ integers $a_1,a_2,\ldots,a_n$ ($1\leq i\leq n$, $-10^9\leq a_i\leq 10^9$).

The third line contains an integer $q$ ($1\leq q\leq 10^5$).

Each of the next $q$ lines contains three integers $l,r,x$ ($1\leq l\leq r\leq n,-10^9\leq x\leq 10^9$), desribing the next change.

Output Format: Print $q+1$ lines.

On the $i$-th ($1 \leq i \leq q+1$) line, print the answer to the problem for the sequence after $i-1$ changes.

Note: In the first test:

• The initial sequence $a = (2, -1, 7, 3)$. Two sequences $b=(-3,-3,5,5),c=(5,2,2,-2)$ is a possible choice.
• After the first change $a = (2, -4, 4, 0)$. Two sequences $b=(-3,-3,5,5),c=(5,-1,-1,-5)$ is a possible choice.
• After the second change $a = (2, -4, 6, 2)$. Two sequences $b=(-4,-4,6,6),c=(6,0,0,-4)$ is a possible choice.