
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  $f(x)$  be a real valued function. Recall that if the inverse function  $f^{-1}(x)$  exists, then  $f^{-1}(x)$  satisfies  $f(f^{-1}(x))=f^{-1}(f(x))=x$ . Given that the inverse of the function  $f(x)=x^3-12x^2+48x-60$  exists, find all real  $a$  that satisfy  $f(a)=f^{-1}(a)$ .

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