A16340 | Freedom of Choice

Let's define the anti-beauty of a multiset $\{b_1, b_2, \ldots, b_{len}\}$ as the number of occurrences of the number $len$ in the multiset.

You are given $m$ multisets, where the $i$ -th multiset contains $n_i$ distinct elements, specifically: $c_{i, 1}$ copies of the number $a_{i,1}$ , $c_{i, 2}$ copies of the number $a_{i,2}, \ldots, c_{i, n_i}$ copies of the number $a_{i, n_i}$ . It is guaranteed that $a_{i, 1} < a_{i, 2} < \ldots < a_{i, n_i}$ . You are also given numbers $l_1, l_2, \ldots, l_m$ and $r_1, r_2, \ldots, r_m$ such that $1 \le l_i \le r_i \le c_{i, 1} + \ldots + c_{i, n_i}$ .

Let's create a multiset $X$ , initially empty. Then, for each $i$ from $1$ to $m$ , you must perform the following action exactly once:

1. Choose some $v_i$ such that $l_i \le v_i \le r_i$
2. Choose any $v_i$ numbers from the $i$ -th multiset and add them to the multiset $X$ .

You need to choose $v_1, \ldots, v_m$ and the added numbers in such a way that the resulting multiset $X$ has the minimum possible anti-beauty.

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

The first line of each test case contains a single integer $m$ ( $1 \le m \le 10^5$ ) — the number of given multisets.

Then, for each $i$ from $1$ to $m$ , a data block consisting of three lines is entered.

The first line of each block contains three integers $n_i, l_i, r_i$ ( $1 \le n_i \le 10^5, 1 \le l_i \le r_i \le c_{i, 1} + \ldots + c_{i, n_i} \le 10^{17}$ ) — the number of distinct numbers in the $i$ -th multiset and the limits on the number of elements to be added to $X$ from the $i$ -th multiset.

The second line of the block contains $n_i$ integers $a_{i, 1}, \ldots, a_{i, n_i}$ ( $1 \le a_{i, 1} < \ldots < a_{i, n_i} \le 10^{17}$ ) — the distinct elements of the $i$ -th multiset.

The third line of the block contains $n_i$ integers $c_{i, 1}, \ldots, c_{i, n_i}$ ( $1 \le c_{i, j} \le 10^{12}$ ) — the number of copies of the elements in the $i$ -th multiset.

It is guaranteed that the sum of the values of $m$ for all test cases does not exceed $10^5$ , and also the sum of $n_i$ for all blocks of all test cases does not exceed $10^5$ .

For each test case, output the minimum possible anti-beauty of the multiset $X$ that you can achieve.