
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 $x$ 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 a line 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 line intersects curve $C$ at points $M$ and $N$.

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

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

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