
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
Elective 4-4: Coordinate System and Parametric Equations

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

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

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

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