
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 parametric equation of curve $C$ is $\begin{cases}x=3\cos \alpha \\ y=\sin \alpha\end{cases} (\alpha$ is the parameter$)$, in the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\rho\sin (\theta- \dfrac{\pi}{4})= \sqrt{2}$.  
$(1)$ Find the general equation of $C$ and the inclination angle of $l$;  
$(2)$ Let point $P(0,2)$, $l$ and $C$ intersect at points $A$ and $B$, find $|PA|+|PB|$.

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