
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]


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Given the function $f(x) = x^3 + ax^2 + bx + c$, if $f(x_1) = x_1$ and $f(x)$ has extreme points $x_1, x_2$, determine the number of distinct real roots of the equation $3(f(x))^2 + 2af(x) + b = 0$.

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