Tree Embedding

bobo has a tree with $n$ vertices. bobo wants to assign an $m$-dimensional vector $\mathbf{p}(v)$ to each vertex $v$ so that for all vertices $a$ and $b$, $\mathrm{dist}(a, b) = \langle \mathbf{p}(a), \mathbf{p}(b) \rangle$.

$\mathrm{dist}(a, b)$ is the length of the shortest path between $a$ and $b$. For vectors $\mathbf{u} = (u_1, \dots, u_m)$ and $\mathbf{v} = (v_1, \dots, v_m)$, the inner product is

$\langle \mathbf{u}, \mathbf{v} \rangle = \max\{|u_1 - v_1|, |u_2 - v_2|, \dots, |u_m - v_m|\}.$

The first line contains an integer $n$ ($2 \leq n \leq 1000$).

Vertices are numbered $1, 2, \dots, n$.

Each of the following $n - 1$ lines contains three integers $a_i$, $b_i$, and $c_i$, meaning there is an edge between vertices $a_i$ and $b_i$ with length $c_i$ ($1 \leq a_i, b_i \leq n$, $1 \leq c_i \leq 100000$).

The first line contains an integer $m$, the dimension of the vectors ($1 \leq m \leq 16$).

Each of the following $n$ lines contains $m$ integers, the vector $\mathbf{p}(i)$. Each coordinate must be in $[-10^9, 10^9]$.

Any valid solution is accepted.