
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 \( f(x) = x^3 + 3x + 1 \), where \( x \) is a real number. Given that the inverse function of \( f \) exists and is given by
\[ f^{-1}(x) = \left( \frac{x - a + \sqrt{x^2 - bx + c}}{2} \right)^{1/3} + \left( \frac{x - a - \sqrt{x^2 - bx + c}}{2} \right)^{1/3} \]
where \( a \), \( b \), and \( c \) are positive constants, find the value of \( a + 10b + 100c \).

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