
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 a parabola $C: y^{2} = 2px (p > 0)$, a point $P(3, t)$ on the parabola is at a distance of $4$ from the focus $F$.  
$(1)$ Find the equation of the parabola;  
$(2)$ A line $l$ passing through the point $(4, 0)$ intersects the parabola $C$ at points $A$ and $B$, with $M(-4, 0)$. If the slopes of lines $AM$ and $BM$ are $k_{1}$ and $k_{2}$ respectively, find the minimum value of $k_{1} \cdot k_{2}$.

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