
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 the line $l: \begin{cases} x=a+t\sin a \\ y=b+t\cos a \end{cases} $, where $t$ is a parameter.

(1) When $a=\frac{\pi }{3}$, find the slope of line $l$;

(2) If $P(a,b)$ is a point inside the circle $O:x^{2}+y^{2}=4$, and $l$ intersects with circle $O$ at points $A$ and $B$, such that $|PA|$, $|OP|$, and $|PB|$ form a geometric sequence, find the trajectory equation of the moving point $P$.

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