You are given a non negative integer k. There is an infinite staircase with stairs numbered from $0$ to infinity. Alice starts on stair $1$ and has a variable jump initialized to $0$.

Alice can perform two types of operations:

1. Move down to stair $i - 1$, where $i$ is her current stair. This operation cannot be performed consecutively or when Alice is already at stair $0$.

2. Move up to stair $i + 2^{ ext{jump}}$. Afterwards, increment jump by $1$.

Return the total number of distinct ways Alice can reach stair k.

Note: A way is defined by the sequence of operations performed to reach stair k. Reaching stair k multiple times across different points in an execution path counts as a distinct way at each arrival.

Constraints:

• $0 \leq$ k $\leq 10^9$

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.