
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
Given that point $F$ is the focus of the parabola $y^{2}=4x$, and $A$, $B$, $C$ are three points on the parabola, and $O$ is the coordinate origin. If $F$ is the centroid of $\triangle ABC$, and the areas of $\triangle OFA$, $\triangle OFB$, and $\triangle OFC$ are respectively $S_{1}$, $S_{2}$, and $S_{3}$, then $S_{1}^{2}+S_{2}^{2}+S_{3}^{2}=$ ___.

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