
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 the focus of the parabola $y^2=2x$, and let $A$, $B$, $C$ be three points on the parabola. If $F$ is the centroid of $\triangle ABC$, then calculate the value of $|\overrightarrow{FA}|+|\overrightarrow{FB}|+|\overrightarrow{FC}|$.

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