Find path in implicit Fibonacci tree

You are given a special family of binary trees called Fibonacci trees. The k‑th order Fibonacci tree T(k) is defined recursively:

• T(1) is a single node.
• T(2) is a single node.
• For k ≥ 3 , T(k) is a tree whose root has:
  • left subtree T(k − 1) and
  • right subtree T(k − 2) .

Let size(k) be the number of nodes in T(k). You can compute size(k) from the definition above:

• size(1) = 1
• size(2) = 1
• size(k) = 1 + size(k − 1) + size(k − 2) for k ≥ 3

Now, imagine we number the nodes of T(k) in preorder (root, then left subtree, then right subtree), using 1‑based indices from 1 to size(k).

You are given:

• An integer k (the order of the Fibonacci tree), with 1 ≤ k ≤ 60 .
• Two integers a and b , such that 1 ≤ a, b ≤ size(k) ; these are indices of two nodes in the preorder numbering of T(k) .

The tree can be very large for big k, so you must not construct it explicitly in memory.

Task

Design and implement a function:

List<Long> pathInFibonacciTree(long k, long a, long b)

that returns the sequence of node indices (in preorder numbering) along the simple path from node a to node b in T(k).

• The path should start with a and end with b .
• Indices in the returned list should be the preorder indices in T(k) .

Constraints and requirements:

• k can be as large as 60 (or similar), so size(k) may be large (up to around 10^12). Use 64‑bit integers for sizes and indices.
• You must treat the tree implicitly :
  • Use the recursive structure and the sizes of subtrees to navigate.
  • You may not allocate O(size(k)) memory or explicitly build all nodes.
• Aim for an algorithm with time complexity polylogarithmic or at worst O(h) , where h is the height of the tree (proportional to k ), i.e., O(k) or similar.

Hints / clarifications

• Because of the preorder layout, for T(k) :
  • The root always has index 1 in its own tree.
  • The left subtree T(k − 1) occupies indices [2, 1 + size(k − 1)] in T(k) .
  • The right subtree T(k − 2) occupies the following range after that.
• You may find it helpful to implement helper functions that, given (k, indexRange, nodeIndex) , determine whether the node lies in the left subtree, right subtree, or is the root.

Design your algorithm and helper functions so that you can:

1. Compute the path from each node up to the root (in terms of indices) without building the tree.
2. Combine these paths to construct the path from a to b via their lowest common ancestor (LCA).

Return the final path as a list of preorder indices.