P9877 [EC Final 2021] Vacation

Prof. Pang has an annual leave of $c$ days and he wants to go on vacation. Now there are $n$ days in a year. Prof. Pang can gain $a_i$ happiness if he rests on the $i$-th day. The values of happiness, $a_i$, may be negative. Prof. Pang wants you to do $m$ operations: - $1~x~y$, change the happiness of the $x$-th day to $y$. - $2~l~r$, Prof. Pang wants to find a period of vacation in $[l, r]$. He wants to rest for several (possibly $0$) days in a row and gain as much happiness as possible. However, he only has $c$ days off, thus he can rest for no more than $c$ consecutive days in $[l,r]$. That means he wants to find $$\max\left(\max_{l \leq l' \leq r' \leq r\atop r'-l'+1\leq c} ~~ \left(\sum_{i=l'} ^{r'} a_i\right), 0\right).$$

The first line contains three integers $n, m, c (1\leq n\leq 2\times 10^5, 1\leq m \leq 5\times 10^5, 1\leq c\leq n)$ indicating the number of days in a year, the number of operations, and Prof. Pang's annual leave days. The next line contains $n$ integers $a_1, a_2, \dots, a_n(-10^9 \leq a_i\leq 10^9)$ indicating the values of happiness of every day. The next $m$ lines are the $m$ operations in the format described above. It is guaranteed that $1\leq x\leq n, -10^9\leq y\leq 10^9, 1\leq l\leq r \leq n$.

For each operation of the second type, print the answer.