
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 $l$, which contains the median from the side $BC$, is $mx+ny-m=0$ ($m, n$ are constants and $m\neq 0$), and the areas of $\triangle OFA$, $\triangle OFB$, and $\triangle OFC$ are 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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