
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 \[ f(x) = \left\{
\begin{array}{cl}
\frac{x}{55} & \text{ if }x\text{ is a multiple of 5 and 11}, \\
5x & \text{ if }x\text{ is only a multiple of 11}, \\
11x & \text{ if }x\text{ is only a multiple of 5}, \\
x+5 & \text{ if }x\text{ is not a multiple of 5 or 11}.
\end{array}
\right. \]
If \( f^a(x) \) means the function is nested \(a\) times (for example, \(f^2(x) = f(f(x))\)), what is the smallest value of \( a \) greater than 1 that satisfies \( f(1) = f^a(1) \)?

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