Description:
Lunchbox has a tree of size $$$n$$$ rooted at node $$$1$$$. Each node is then assigned a value. Lunchbox considers the tree to be beautiful if each value is distinct and ranges from $$$1$$$ to $$$n$$$. In addition, a beautiful tree must also satisfy $$$m$$$ requirements of $$$2$$$ types:
- "1 a b c" — The node with the smallest value on the path between nodes $$$a$$$ and $$$b$$$ must be located at $$$c$$$.
- "2 a b c" — The node with the largest value on the path between nodes $$$a$$$ and $$$b$$$ must be located at $$$c$$$.
Now, you must assign values to each node such that the resulting tree is beautiful. If it is impossible to do so, output $$$-1$$$.
Input Format:
The first line contains two integers $$$n$$$ and $$$m$$$ ($$$2 \le n, m \le 2 \cdot 10^5$$$).
The next $$$n - 1$$$ lines contain two integers $$$u$$$ and $$$v$$$ ($$$1 \le u, v \le n, u \ne v$$$) — denoting an edge between nodes $$$u$$$ and $$$v$$$. It is guaranteed that the given edges form a tree.
The next $$$m$$$ lines each contain four integers $$$t$$$, $$$a$$$, $$$b$$$, and $$$c$$$ ($$$t \in \{1,2\}$$$, $$$1 \le a, b, c \le n$$$). It is guaranteed that node $$$c$$$ is on the path between nodes $$$a$$$ and $$$b$$$.
Output Format:
If it is impossible to assign values such that the tree is beautiful, output $$$-1$$$. Otherwise, output $$$n$$$ integers, the $$$i$$$-th of which denotes the value of node $$$i$$$.
Note:
None