
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
In the plane rectangular coordinate system $xOy$, the equation of curve $C$ is $\left(x-3\right)^{2}+y^{2}=4$. The line $l$ passes through the point $P\left(3,1\right)$ with an inclination angle of $\alpha$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive $x$-axis as the polar axis.<br/>$(1)$ Write down the parametric equation of line $l$ (expressed in terms of the coordinates of point $P$ and $\alpha$) and the polar coordinate equation of curve $C$;<br/>$(2)$ Suppose the line $l$ intersects curve $C$ at points $A$ and $B$, find the minimum value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

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