A15567 | Xorcerer's Stones

Misha had been banned from playing chess for good since he was accused of cheating with an engine. Therefore, he retired and decided to become a xorcerer.

One day, while taking a walk in a park, Misha came across a rooted tree with nodes numbered from $1$ to $n$ . The root of the tree is node $1$ .

For each $1\le i\le n$ , node $i$ contains $a_i$ stones in it. Misha has recently learned a new spell in his xorcery class and wants to test it out. A spell consists of:

- Choose some node $i$ ( $1 \leq i \leq n$ ).
- Calculate the [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation#XOR) $x$ of all $a_j$ such that node $j$ is in the subtree of $i$ ( $i$ belongs to its own subtree).
- Set $a_j$ equal to $x$ for all nodes $j$ in the subtree of $i$ .

Misha can perform at most $2n$ spells and he wants to remove all stones from the tree. More formally, he wants $a_i=0$ to hold for each $1\leq i \leq n$ . Can you help him perform the spells?

A tree with $n$ nodes is a connected acyclic graph which contains $n-1$ edges. The subtree of node $i$ is the set of all nodes $j$ such that $i$ lies on the simple path from $1$ (the root) to $j$ . We consider $i$ to be contained in its own subtree.

The first line contains a single integer $n$ ( $2 \leq n \leq 2\cdot 10^5$ ) — the size of the tree

The second line contains an array of integers $a_1,a_2,\ldots, a_n$ ( $0 \leq a_i \leq 31$ ), describing the number of stones in each node initially.

The third line contains an array of integers $p_2,p_3,\ldots, p_n$ ( $1 \leq p_i \leq i-1$ ), where $p_i$ means that there is an edge connecting $p_i$ and $i$ .

If there is not a valid sequence of spells, output $-1$ .

Otherwise, output a single integer $q$ ( $0 \leq q \leq 2n$ ) in the first line — the number of performed spells.

In the second line output a sequence of integers $v_1,v_2,\ldots,v_q$ ( $1 \leq v_i \leq n$ ) — the $i$ -th spell will be performed on the subtree of node $v_i$ . Please note that order matters.

If multiple solutions exist, output any. You don't have to minimize the number of operations.