
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 $(xOy)$, the parameteric equations of line $l$ are given by $\begin{cases} x=\sqrt{3}t \\ y=1+t \end{cases} (t \text{ is the parameter})$. Establish a polar coordinate system with the origin $O$ as the pole and the non-negative $x$-axis as the polar axis. The polar equation of the curve $C$ is $\rho^2\cos2\theta=1$. The line $l$ intersects the curve $C$ at points $A$ and $B$.

(I) Find the length of $|AB|$;
(II) If point $P$ has polar coordinates $(1, \frac{\pi}{2})$, find the distance between the midpoint $M$ of $AB$ and $P$.

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