
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
Suppose the sequence $\{a\_n\}$ satisfies the following condition: for any $n\in\mathbb{N}^*$, there are only finitely many positive integers $m$ such that $a\_m < n$ holds. Let $b\_n$ denote the number of such $m$'s. This gives rise to a new sequence $\{b\_n\}$. For example, if the sequence $\{a\_n\}$ is $1$, $2$, $3$, $...$, $n...$, then the sequence $\{b\_n\}$ is $0$, $1$, $2$, $...$, $n-1$, $...$ Now, it is known that the sequence $\{a\_n\}$ is a geometric sequence with $a\_2=2$, $a\_5=16$. Find the number of positive integers $i$ such that $b\_i=2016$ in the sequence $\{b\_n\}$.

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