
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 $f(x) = (x-1)^3 - ax + b$, $x \in \mathbb{R}$, where $a, b \in \mathbb{R}$.  
$(1)$ Find the monotonic intervals of $f(x)$.  
$(2)$ If $f(x)$ has an extremum point $x_{0}$ and $f(x_{1}) = f(x_{0})$, where $x_{1} \neq x_{0}$, find the value of $x_{1} + 2x_{0}$.

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