
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 the three vertices of $\triangle ABC$ are all on the parabola $y^{2}=2px(p > 0)$, and $O$ is the vertex of the parabola, the focus $F$ of the parabola satisfies $\overrightarrow{FA}+ \overrightarrow{FB}+ \overrightarrow{FC}= \overrightarrow{0}$. If the equation of the line $l$ containing the median on side $BC$ is $mx+ny-m=0$ ($m,n$ are constants and $m\neq 0$), and let the areas of $\triangle OFA$, $\triangle OFB$, and $\triangle OFC$ be denoted as $S_{1}$, $S_{2}$, and $S_{3}$ respectively, then the value of $S_{1}^{2}+S_{2}^{2}+S_{3}^{2}$ is \_\_\_\_\_\_.

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