CF1860F.Evaluate RBS

You are given $2n$ tuples of values $(a, b, c)$ , where $a$ and $b$ are positive integers and $c$ is a bracket '(' or ')'. Exactly $n$ tuples have $c = $ '(' and the other $n$ tuples have $c =$ ')'.

You are asked to choose two positive values $x$ and $y$ ( $x > 0$ and $y > 0$ ; not necessarily integers) and sort the tuples in the increasing value of $a \cdot x + b \cdot y$ . If several tuples have the same value, you can place them in any order among themselves.

Is it possible to choose such $x$ and $y$ that taking brackets $c$ from the tuples in the resulting order produces a regular bracket sequence?

A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters "1" and "+" between the original characters of the sequence.

The first line contains a single integer $t$ ( $1 \le t \le 1500$ ) — the number of testcases.

The first line of each testcase contains a single integer $n$ ( $1 \le n \le 1500$ ).

The $i$ -th of the next $2n$ lines contains two integers $a_i$ and $b_i$ ( $1 \le a_i, b_i \le 10^6$ ) and a character $c_i$ ( $c_i = $ '(' or $c_i = $ ')'). Exactly $n$ tuples have $c_i = $ '(' and the other $n$ tuples have $c_i =$ ')'.

The sum of $n$ over all testcases doesn't exceed $1500$ .

For each testcase, print "YES" if it's possible to choose such $x$ and $y$ that taking brackets $c$ from the tuples in the resulting order produces a regular bracket sequence. Otherwise, print "NO".

In the first testcase, you can choose $x = 10, y = 0.1$ . The values for tuples will be $10 \cdot 1 + 0.1 \cdot 4 = 10.4$ and $10 \cdot 2 + 0.1 \cdot 3 = 20.3$ . Thus, the first tuple will go first, then the second one, making the bracket sequence "()", which is regular.

In the second testcase, you can't choose positive $x$ and $y$ such that the opening brackets gets assigned a value less or equal to the value of the closing bracket.

In the fourth testcase, you can choose $x = 0.6, y = 0.55$ . The bracket sequence is "(()(()))".