
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Define the sequence $\left\{a_{n}\right\}$ such that:
\[
\begin{array}{l}
a_{1}=0, \\
a_{n}=a_{\left[\frac{n}{2}\right]}+(-1)^{\frac{n(n+1)}{2}} \quad \text{for } n=2,3, \cdots.
\end{array}
\]

For each non-negative integer $k$, find the number of indices $n$ that satisfy $2^{k} \leqslant n < 2^{k+1}$ and $a_{n}=0$.

Present the answer in LaTex format: \boxed{Your answer}
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