CF1654D.Potion Brewing Class

Alice's potion making professor gave the following assignment to his students: brew a potion using $n$ ingredients, such that the proportion of ingredient $i$ in the final potion is $r_i > 0$ (and $r_1 + r_2 + \cdots + r_n = 1$ ).

He forgot the recipe, and now all he remembers is a set of $n-1$ facts of the form, "ingredients $i$ and $j$ should have a ratio of $x$ to $y$ " (i.e., if $a_i$ and $a_j$ are the amounts of ingredient $i$ and $j$ in the potion respectively, then it must hold $a_i/a_j = x/y$ ), where $x$ and $y$ are positive integers. However, it is guaranteed that the set of facts he remembers is sufficient to uniquely determine the original values $r_i$ .

He decided that he will allow the students to pass the class as long as they submit a potion which satisfies all of the $n-1$ requirements (there may be many such satisfactory potions), and contains a positive integer amount of each ingredient.

Find the minimum total amount of ingredients needed to make a potion which passes the class. As the result can be very large, you should print the answer modulo $998\,244\,353$ .

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

The first line of each test case contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ).

Each of the next $n-1$ lines contains four integers $i, j, x, y$ ( $1 \le i, j \le n$ , $i\not=j$ , $1\le x, y \le n$ ) — ingredients $i$ and $j$ should have a ratio of $x$ to $y$ . It is guaranteed that the set of facts is sufficient to uniquely determine the original values $r_i$ .

It is also guaranteed that the sum of $n$ for all test cases does not exceed $2 \cdot 10^5$ .

For each test case, print the minimum total amount of ingredients needed to make a potion which passes the class, modulo $998\,244\,353$ .

In the first test case, the minimum total amount of ingredients is $69$ . In fact, the amounts of ingredients $1, 2, 3, 4$ of a valid potion are $16, 12, 9, 32$ , respectively. The potion is valid because

• Ingredients $3$ and $2$ have a ratio of $9 : 12 = 3 : 4$ ;
• Ingredients $1$ and $2$ have a ratio of $16 : 12 = 4 : 3$ ;
• Ingredients $1$ and $4$ have a ratio of $16 : 32 = 2 : 4$ .

In the second test case, the amounts of ingredients $1, 2, 3, 4, 5, 6, 7, 8$ in the potion that minimizes the total amount of ingredients are $60, 60, 24, 48, 32, 60, 45, 30$ .