
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 line $l$ passing through point $M\left(\frac{p}{2}, 0\right)$ intersects the parabola $y^2=2px$ ($p>0$) at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = -3$, where $O$ is the origin of the coordinate system.  
(1) Find the value of $p$;  
(2) When $|AM|+4|BM|$ is minimized, find the equation of line $l$.

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