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

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

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

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