
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
Given the quadratic function $f(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers and the inverse function is given by $f^{-1}(x) = cx^2 + bx + a$. Determine the values of $a$, $b$, and $c$ such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

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