
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 parametric equation of curve $C\_1$ as $\begin{cases} x=2\cos \theta \ y=\sqrt{3}\sin \theta \end{cases}(\theta \text{ is the parameter})$, and curve $C\_2$ has a polar coordinate equation of $\rho=2$ with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.

(1) Write the Cartesian equation for $C\_1$ and the polar coordinate equation for $C\_2$.
(2) Let $M$ and $N$ be the upper and lower vertices of curve $C\_1$, respectively. $P$ is any point on curve $C\_2$. Find the maximum value of $|PM|+|PN|$.

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