
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 a Cartesian coordinate system with its origin $O$ as the pole, the positive semi-axis of the $x$-axis as the polar axis, and both coordinate systems having the same unit length, the parameteric equations of line $l$ are $\begin{cases} x=\frac{\sqrt{2}}{2}t \\ y=3+\frac{\sqrt{2}}{2}t \end{cases} (t \text{ is the parameter})$, and the polar equation of curve $C$ is $\rho \cos^{2}\theta =2\sin \theta$.

(I) Write the Cartesian equation of line $l$ and the polar equation of curve $C$.
(II) Suppose line $l$ intersects curve $C$ at points $A$ and $B$, and the polar coordinates of point $P$ are $(3, \frac{\pi}{2})$. Find the value of $|PA| + |PB|$.

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