Interval Multiset Maintenance and Exact Cover Query

You are given two integers \(n\) and \(m\). We maintain a multiset (i.e. a set that allows duplicate intervals) of intervals on the number line [1, n) (with the half-open interval notation) that is initially empty. You need to process \(m\) operations. There are three types of operations:

1. Insertion: Given two integers \(l\) and \(r\) with \(1 \le l < r \le n\) insert the interval \([l, r)\) into the multiset.
2. Deletion: Given an integer \(t\), remove the interval that was inserted in the \(t\text{-th}\) insertion operation. It is guaranteed that this interval has not been removed yet.
3. Query: Given an interval \([l, r)\), determine whether there exists a subset of the current multiset of intervals such that the union of the chosen intervals is exactly \([l, r)\). Note that if an interval in the subset extends outside \([l, r)\), then the union will not be exactly \([l, r)\). Thus, only the intervals completely contained in \([l, r)\) can be chosen.

For the query, you can use a greedy strategy: filter the active intervals that are completely within \([l, r)\), sort them by their starting point and then try to cover the entire interval \([l, r)\) without gaps. If you can cover from \(l\) to \(r\) exactly, output YES; otherwise, output NO.

inputFormat

The first line contains two integers \(n\) and \(m\) (\(1 \le n \le 10^9\); \(1 \le m \le 10^5\)), representing the length of the number line and the number of operations respectively.

Each of the next \(m\) lines describes an operation in one of the following formats:

• For an insertion operation: 1 l r (inserts the interval \([l, r)\)).
• For a deletion operation: 2 t (deletes the interval inserted in the \(t\)-th insertion operation).
• For a query operation: 3 l r (queries whether there exists a subset of intervals whose union is exactly \([l, r)\)).

outputFormat

For each query operation (operation type 3), output a line containing either YES if there exists a subset of active intervals whose union is exactly \([l, r)\), or NO otherwise.

sample

10 6
1 1 3
1 3 5
1 5 10
3 1 10
2 2
3 1 10

YES
NO