
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 $C:y^{2}=2px (p > 0)$ with a point $P(3, t)$ on it, the distance from point $P$ to the focus $F$ is $4$.

(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 $B$. Let $M(-4,0)$, and 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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