
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
(1) Given a moving circle passes through a fixed point $A(4,0)$, and cuts a chord $MN$ of length $8$ on the $y$-axis. Find the equation of the trajectory $C$ of the center of the moving circle;

(2) In the Cartesian coordinate system $xOy$, a line $l$ intersects the parabola $y^{2}=4x$ at two distinct points $A$ and $B$. If the line $l$ passes through the focus of the parabola, find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$;

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