
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 non-negative half-axis of the $x$ axis as the polar axis, the curve $C$: $\rho\sin^2\theta=4\cos\theta$, and the parametric equation of line $l$: 
$$
\begin{cases} 
x=-2+ \frac{\sqrt{2}}{2}t \\
y=-4+ \frac{\sqrt{2}}{2}t
\end{cases}
(t \text{ is the parameter})
$$
The two curves intersect at points $M$ and $N$.

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

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

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