
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
Let line $l$ intersect parabola $\Omega: y^{2}=4x$ at two distinct points $A$ and $B$, and $O$ be the origin.  
(1) Calculate the distance from the focus of the parabola $\Omega$ to its directrix;  
(2) If the line $l$ is also tangent to the circle $C: (x-5)^{2}+y^{2}=16$ at point $M$, and $M$ is the midpoint of segment $AB$, find the equation of line $l$;  
(3) If $ \overrightarrow{OA} \cdot \overrightarrow{OB}=0$, and point $Q$ is on segment $AB$ such that $OQ \perp AB$, find the equation of the trajectory of point $Q$.

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