
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
Consider the sequence of numbers defined recursively by $s_1=2$ and for $n>1$ by $s_n=2+s_{n/4}$ when $n$ is a multiple of 4, by $s_n=\frac{1}{s_{n-1}}$ when $n$ is odd, and by $s_n=s_{n-1}+1$ when $n$ is even but not a multiple of 4. Given that $s_n=\frac{5}{36}$, find $n$.

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