Background

Problem Description

Hoshino Ai gives you a sequence $a_1, a_2, \ldots, a_n$ of length $n$. Each number is between $0$ and $v - 1$. There are $m$ operations.

Operation $1$: Each time, ask whether it is possible to choose two non-empty sets of indices $X, Y$ in an interval such that:

1. $X$ and $Y$ are disjoint.
2. For an element $i \in X$, its contribution to set $X$ is $a_i + 1$. The total contribution of elements in $X$ must be equal to the total contribution of elements in $Y$. If such two sets can be chosen, output Yuno; otherwise output Yuki.

Operation $2$: Modify the numbers in an interval $[l, r]$ so that for all $l \leq i \leq r$, $a_i = a_i^3 \bmod v$, i.e., take the cube modulo $v$ on the interval.

Input Format

The first line contains three integers $n, m, v$, with meanings as described above.

The next line contains $n$ integers, representing the sequence $a$.

Then follow $m$ lines, each containing three numbers $opt, l, r$, indicating whether the operation type is $1$ or $2$, and the operated interval is $[l, r]$.

Output Format

For each query, output one line containing a string Yuno or Yuki, indicating whether such two sets can be chosen.

20 20 152
3 26 133 54 79 81 72 109 66 91 82 100 35 23 104 17 51 114 12 58
2 1 17
2 6 12
1 1 12
2 3 5
2 11 11
2 7 19
2 6 15
1 5 12
1 1 9
1 10 19
2 3 19
2 6 20
2 1 13
2 1 15
2 1 9
1 1 1
2 1 7
2 7 19
2 6 19
2 3 6

Yuno
Yuno
Yuno
Yuno
Yuki

Hint

Idea: nzhtl1477, Solution: nzhtl1477, Code: nzhtl1477, Data: nzhtl1477.

Constraints: For $100\%$ of the testdata, $1 \leq n, m \leq 10^5$, $1 \leq v \leq 1000$. The testdata has no subtasks or grading by difficulty.

Translated by ChatGPT 5