
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 rectangular coordinate system $(xOy)$, let $l$ be a line with an inclination angle of $\alpha$ and $C$ be a curve, both parameterized as follows:

$l: \begin{cases} x=2+t\cos\alpha \\ y=\sqrt{3}+t\sin\alpha \end{cases} (t \text{ is a parameter})$

$C: \begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases} (\theta \text{ is a parameter})$

The line $l$ and the curve $C$ intersect at two distinct points $A$ and $B$.

(1) If $\alpha=\frac{\pi}{3}$, find the coordinates of the midpoint $M$ of the segment $AB$.

(2) If $|PA|\cdot|PB|=|OP|^2$, where $P(2,\sqrt{3})$, find the slope of the line $l$.

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