
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 \( f \) be a monic cubic polynomial satisfying \( f(x) + f(-x) = 0 \) for all real numbers \( x \). For all real numbers \( y \), define \( g(y) \) to be the number of distinct real solutions \( x \) to the equation \( f(f(x)) = y \). Suppose that the set of possible values of \( g(y) \) over all real numbers \( y \) is exactly \(\{1,5,9\}\). Compute the sum of all possible values of \( f(10) \).

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