
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 $P(2,1)$, a line $l$ is defined by $\left\{\begin{array}{l}x=2+t\cos\alpha\\ y=1+t\sin\alpha\end{array}\right.$ ($t$ is a parameter), where $\alpha$ is the inclination angle of $l$. Line $l$ intersects the positive $x$-axis and positive $y$-axis at points $A$ and $B$ respectively, such that $|PA|\cdot |PB|=4$. <br/>$(1)$ Find the value of $\alpha$; <br/>$(2)$ Establish the polar coordinate equation of $l$ with the origin as the pole and the positive $x$-axis as the polar axis.

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