
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 the ellipse $C$ is 
$$
\begin{cases} 
x= \sqrt {3}\cos \varphi \\ 
y= \sqrt {2}\sin \varphi
\end{cases}
(\varphi \text{ is the parameter})
$$
Taking the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\rho\cos \theta+\rho\sin \theta=1$.

$(1)$ Find the polar equation of the ellipse $C$ and the parametric equation of the line $l$;  
$(2)$ If the polar coordinates of point $P$ are $(1, \frac {\pi}{2})$, and line $l$ intersects ellipse $C$ at points $A$ and $B$, find the value of $|PA|+|PB|$.

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