
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 that $O$ is the coordinate origin and $M(s,t)$ is a moving point.

(I) When $s=0$, $t=4 \sqrt {3}$, the line $l$ passing through point $M$ is tangent to the circle $C$: $x^{2}+y^{2}-8x=0$. Find the equation of the line $l$.

(II) When $s=2$, $t > 0$, find the equation of the circle with $OM$ as the diameter and the chord length of $2$ intercepted by the line $3x-4y-5=0$.

(III) When $s=2$, $t > 0$, let $A(1,0)$. The perpendicular line to $OM$ passing through point $A$ intersects the circle with $OM$ as the diameter at point $N$, and the foot of the perpendicular is $H$. Determine whether the length of the line segment $ON$ is a constant value. If it is, find this value; if not, explain the reason.

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