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 "(()(()))".