Path Between Two Preorder-Numbered Nodes of an Implicit Fibonacci Tree

Fibonacci trees are defined recursively. `T(1)` and `T(2)` are single-node trees. For `k >= 3`, `T(k)` has a root whose left subtree is `T(k - 1)` and whose right subtree is `T(k - 2)`. Let `size(k)` be the number of nodes in `T(k)`: `size(1) = 1`, `size(2) = 1`, and `size(k) = 1 + size(k - 1) + size(k - 2)`. Nodes are numbered 1, 2, 3, ... in the order a preorder traversal (root, then left subtree, then right subtree) visits them, so the root is node 1. Given the order `k` and two nodes `a` and `b`, identified by their preorder numbers, return the preorder numbers of the nodes on the unique simple path from `a` to `b`. You must not construct the tree explicitly; for large `k` it has trillions of nodes. ### Function Signature ```python def path_in_fibonacci_tree(k: int, a: int, b: int) -> list[int]: ``` ### Rules - The returned list starts with `a`, ends with `b`, and lists the nodes in the order the path visits them. The original interface accepted the nodes in any order; this console fixes the order so the answer is unique. - If `a == b`, return `[a]`. - Each node on the path appears exactly once. ### Constraints - `1 <= k <= 60` - `1 <= a, b <= size(k)` - `size(60) = 3,096,017,511,839`, which exceeds `2^31 - 1` but stays below `2^53`; use 64-bit integers in fixed-width languages. ### Examples For reference, here is `T(5)` numbered in preorder (the left child is listed first): ```text 1 +-- 2 | +-- 3 | | +-- 4 | | +-- 5 | +-- 6 +-- 7 +-- 8 +-- 9 ``` **Example 1** ```text k = 5, a = 4, b = 9 Output: [4, 3, 2, 1, 7, 9] ``` **Example 2** ```text k = 5, a = 6, b = 5 Output: [6, 2, 3, 5] ``` **Example 3** ```text k = 3, a = 2, b = 2 Output: [2] ```