
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 Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $ \begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases} $ (where $\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $\rho\sin (\theta- \frac {\pi}{4})= \sqrt {2}$.  
$(1)$ Find the implicit equation for $C$ and the inclination angle of $l$.  
$(2)$ Let point $P(0,2)$ be given, and line $l$ intersects curve $C$ at points $A$ and $B$. Find the $|PA|+|PB|$.

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