Mod Graph

You are given an undirected, connected graph with $n$ vertices. Every vertex $v$ has a counter that operates modulo $b_v$. The initial state of that counter is $a_v$. Every time you visit that vertex, it is increased by one (i.e. $a_v \leftarrow (a_v + 1) \bmod b_v$).

You have to process $q$ queries of the following two types:

• "1 $s$": Suppose that you start at vertex $s$, you have to answer whether it is possible to make $a_v = 0$ for all $v$. You are allowed to take an arbitrary walk through $G$, i.e. you can use edges and vertices multiple times. Note that starting the walk at $s$ already counts as visiting $s$, i.e. its counter is increased by one initially.
• "2 $v$ $x$": Update $a_v \leftarrow x$.

The first line contains three integers $n$, $m$, and $q$ ($1 \leq n,q \leq 5 \cdot 10^4$, $0 \leq m \leq 10^5$) --- the number of vertices, edges, and queries, respectively.

The second line contains $n$ integers $a_1, \ldots, a_n$.

The third line contains $n$ integers $b_1, \ldots, b_n$ ($0 \leq a_v < b_v \leq 10^9$).

The following $m$ lines contain the edges of the graph. Each of those $m$ lines contains two integers $u$ and $v$ ($1 \leq u, v \leq n$, $u \neq v$) describing an edge connecting vertices $u$ and $v$. The given graph is connected.

Finally, there are $q$ lines describing the queries. They can be in the two formats described above, i.e.

• "1 $s$": ($1 \leq s \leq n$)
• "2 $v$ $x$": ($1 \leq v \leq n$, $0 \leq x < b_v$)

For each query of type 1, output YES in one line if it is possible to make $a_v = 0$ for all $v$ and NO otherwise.

In the first query, we start at vertex $1$ with counter states $[1, 0]$.

We move along the walk $1 \rightarrow 2 \rightarrow 1 \rightarrow 2 \rightarrow 1 \rightarrow 2$.

The counter states change as follows: $[1, 0] \rightarrow [1, 1] \rightarrow [2, 1] \rightarrow [2, 2] \rightarrow [0, 2] \rightarrow [0, 0]$.