
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 equations of curve $C_1$ as $\begin{cases} x=2\cos \theta \\ y=\sqrt{3}\sin \theta \end{cases}$ ($\theta$ 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 Cartesian equation for $C_2$.
(2) Let $M$ and $N$ be the upper and lower vertices of curve $C_1$, respectively, and $P$ is an arbitrary 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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