
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 $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve $C$ is $\rho\sin^2\theta=4\cos\theta$, and the parametric equation of line $l$ is $\begin{cases} x=-2+ \frac{\sqrt{2}}{2}t, \\ y=-4+ \frac{\sqrt{2}}{2}t \end{cases}$ (where $t$ is the parameter), and the two curves intersect at points $M$ and $N$.
$(1)$ Write the Cartesian coordinate equation of curve $C$ and the general equation of line $l$;

$(2)$ If $P(-2,-4)$, find the value of $|PM|+|PN|$.

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