
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, the parametric equation of curve $C$ is 
$$
\begin{cases} 
& x=\sqrt{5}\cos \alpha \\
& y=\sin \alpha \\
\end{cases} 
(\alpha \text{ is the parameter}).
$$
Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\rho \cos (\theta +\frac{\pi }{4})=\sqrt{2}$. The line $\sqrt{2}l$ intersects curve $C$ at points $A$ and $B$.

$(1)$ Find the standard equation of curve $C$ and the Cartesian coordinate equation of line $l$;

$(2)$ Given point $P(0,-2)$, find the value of $|PA|+|PB|$.

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