
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
Solve the equation \( f^{-1}(g(x))=h(x) \), where \( f^{-1}(x) \) is the inverse function of \( f(x) \), given that \( f(x)=x^{3}+2x^{2}+3x+1 \), \( g(x)=x^{4}-x^{3}+4x^{2}+8x+8 \), \( h(x)=x+1 \).

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