
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
In the positive arithmetic sequence $\{a_n\}$, $a_1$ and $a_4$ are the two roots of the equation $x^2-10x+16=0$. If the sum of the first $5$ terms of the sequence $\{\log_2 a_n\}$ is $S_5$ and $S_5 \in [n, n+1]$ where $n \in \mathbb{Z}$, then $n=$ ______.

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