
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
Consider a parabola $y^2=2px$ ($p>0$) with focus $F$. Let $M$ and $N$ be the intersection points of the parabola and a line through focus $F$. Drop perpendiculars from $M$ and $N$ to the directrix $l$, and denote the feet of these perpendiculars as $M_1$ and $N_1$, respectively.

1. Find $\overrightarrow{FM_1} \cdot \overrightarrow{FN_1}$.
2. Let $S_1$, $S_2$, and $S_3$ be the areas of triangles $\triangle FMM_1$, $\triangle FM_1N_1$, and $\triangle FNN_1$, respectively. Find $\frac{S_2^2}{S_1S_3}$.

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