
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
Consider a continuous function $y=f(x)$ defined on the interval $[a, b]$. If there exists $x_0 \in [a, b]$ such that $f(x_0) = \frac{\int_{a}^{b} f(x) dx}{b-a}$ holds, then determine the number of "mean value points" of the function $f(x) = x^3 + 2x$ on $[-1, 1]$.

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