
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
Let \[g(x) = \left\{
\begin{array}{cl}
\frac{x}{15} & \text{ if }x\text{ is a multiple of 5 and 3}, \\
5x & \text{ if }x\text{ is only a multiple of 3}, \\
3x & \text{ if }x\text{ is only a multiple of 5}, \\
x+5 & \text{ if }x\text{ is not a multiple of 3 or 5}.
\end{array}
\right.\]
If $g^b(x)$ means the function is nested $b$ times (for example, $g^2(x)=g(g(x))$), what is the smallest value of $b$ greater than 1 that satisfies $g(2)=g^b(2)$?

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