
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
On the parabola $y^{2}=4x$, there is a moving point $P\left(s,t\right)$, where $t \gt 0$. A tangent line $l$ is drawn passing through point $P$ on the parabola, and a line $m$ passing through point $P$ is drawn such that $m\bot l$. The intersection of line $m$ and the parabola at another point is $Q$.<br/>$(1)$ When $s=1$, find the equation of the tangent line $l$;<br/>$(2)$ When the intersection point of line $l$ and the axis of the parabola is on the $x$-axis, find the area of triangle $OPQ$ (where point $O$ is the origin);<br/>$(3)$ Find the expression of the line segment $|PQ|$ in terms of $s$, and determine the minimum value of $|PQ|$.

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