
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 $xOy$, there is a line $l$ with slope $1$ passing through a fixed point $(-2,-4)$. A polar coordinate system is established with $O$ as the pole and the non-negative half of the $x$-axis as the polar axis. The curve $C$ is given by the polar equation $\rho\sin^2\theta - 4\cos\theta=0$.
$(1)$ Determine the Cartesian equation of curve $C$ and the parametric equation of line $l$;
$(2)$ The curve $C$ intersects line $l$ at points $M$ and $N$. Given point $P(-2,-4)$, find the value of $|PM|+|PN|$.

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