Centered Block Placement Queries

Process queries on an infinite integer number line while maintaining obstacle coordinates.
[1, x] places an obstacle at coordinate x. That coordinate is guaranteed to be empty.
[2, x, size] checks a centered block whose occupied integer coordinates run from x - size + 1 through x + size - 1, inclusive. Append '1' when the entire range is obstacle-free and '0' otherwise. A check does not place a block.
Return the results of all checks as one binary string.

Function
processBlockQueries(queries: int[][]) → String

Examples
Example 1
queries = [[2,0,3],[1,2],[2,0,3],[2,-3,2]]
return = "101"
The first check covers -2 through 2 and is clear. After inserting at 2, the same range is blocked. The final check covers -4 through -2 and remains clear.
Example 2
queries = [[1,5],[2,5,1],[2,4,1],[2,3,3]]
return = "010"
A size-one check covers only its center. The last range covers 1 through 5 and includes the obstacle.

Constraints
1 <= queries.length <= 100000.
Insertion queries have two integers and check queries have three integers.
-10^9 <= x <= 10^9 and 1 <= size <= 10^9.
Inserted obstacle coordinates are distinct.

The trick is that you only need the nearest obstacle to the range, not every obstacle. Keep obstacles in a sorted structure, find the smallest obstacle >= left (lower bound), and check whether it's <= right. If yes, output 0. Otherwise 1. That's O(log n) per query. In Java, use a TreeSet with ceiling(left). In C++, use std::set with lower_bound. Python has no built-in sorted set, so you'd need a sorted list with bisect (insertion is O(n) but often passes) or an offline approach: read all queries, compress coordinates, and use a Fenwick or segment tree for counts. The common pitfalls are off-by-one on the range (x - size + 1 to x + size - 1, inclusive) and overflow when computing with 10^9 values in a 32-bit int. A size of 1 covers only the center. If you freeze on the data structure choice live, StealthCoder is the hedge, but the ceiling lookup is the whole problem.

StealthCoder is the hedge for the one pattern you didn't drill. It runs invisibly during the screen share.