
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 parabola ${\Omega : y^{2}= 2px(0\lt p\leq 8)}$ with focus ${F}$ and a circle ${C}$: ${(x-3)^{2}+ y^{2}= 1}$, where the tangent line ${FT}$ (with ${T}$ as the point of tangency) is drawn from the focus ${F}$ to the circle ${C}$, and the length of the tangent line ${FT}$ is ${\sqrt{3}}$. <br/>(${}$I${}) Find the equation of the parabola ${C}$; <br/>(${}$II${}) Construct a tangent line ${l}$ to the circle ${C}$: ${(x-3)^{2}+ y^{2}= 1}$, where the line ${l}$ intersects the parabola ${\Omega}$ at points ${A}$ and ${B}$, and find the minimum value of ${\mathrel{|} FA\mathrel{|} \cdot \mathrel{|} FB\mathrel{|} }$.

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