
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 $l$ intersecting the parabola $\Omega :{{y}^{2}}=4x$ at two distinct points $A$ and $B$, with $O$ as the coordinate origin.

(1) Find the distance from the focus of the parabola $\Omega$ to its directrix.

(2) If line $l$ is also tangent to the circle $C:{\left(x-5\right)}^{2}+{y}^{2}=16$ at point $M$, which is the midpoint of segment $AB$, find the equation of line $l$.

(3) If $\overrightarrow{OA}\cdot \overrightarrow{OB}=0$, and a point $Q$ lies on segment $AB$ such that $OQ\bot AB$, find the locus equation of point $Q$.

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