Chain Booking — Max Cascading Reservations

Given a list of listings, each described as `[x, y, r]` (coordinates plus a chain-effect radius), booking one listing auto-books every listing within its radius, which then recursively cascades. Choosing exactly one starting listing, return the maximum number of listings that can end up booked. Reduces to a max-reachable-set over a directed graph built from radius containment; candidates solve it with DFS/BFS from each node (SCC-condensation + DAG DP is the optimization). Requirements Input: a list of listings, each [x, y, r] — (x, y) is the coordinate and r is the chain-effect radius. Example input: [[1, 2, 3], [5, 2, 2], [3, 4, 1]]. Booking a listing automatically books every other listing whose coordinate lies within radius r of it. Each newly booked listing then fires its own radius, cascading recursively. You may book exactly one listing. Return the maximum number of listings that end up booked across all single starting choices. You write your own test cases — the round provides no test harness. Notes Model as a directed graph: add edge A → B when B lies within A's radius (Euclidean distance from A to B ≤ r_A). Edges are directional and not symmetric — each node has its own radius, so A → B does not imply B → A. The answer is the size of the largest reachable set over all starting nodes: run a DFS/BFS from each node with a visited set and take the max. The optimization is to condense strongly-connected components and DP the reachable count over the resulting DAG. Track visited per traversal so a cascade cannot loop forever or double-count on cycles — this is the most common bug. A straightforward per-node DFS is O(V·(V+E)), which is fine for the small inputs given; a working non-optimal solution that passes self-written tests is accepted, with the SCC/DAG speed-up reserved as a follow-up. Preparation Build the radius-containment adjacency list, then write a reusable reachable_count(start) DFS with a visited set; drill the whole thing under 25 minutes including your own test cases. Practice articulating the SCC-condensation + DAG-DP optimization out loud as the "make it faster" answer, even if you code only the brute-force version.