
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
【Elective 4-4: Coordinate System and Parametric Equations】

In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the curve $C: \rho \sin^2 \theta = 2a \cos \theta (a > 0)$, and the line $l$ passing through the point $P(-2, -4)$ has the parametric equations: $\begin{cases}x = -2 + \frac{\sqrt{2}}{2}t \\ y = -4 + \frac{\sqrt{2}}{2}t\end{cases} (t \text{ is a parameter})$. The line $l$ intersects the curve $C$ at points $M$ and $N$.

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

$(2)$ If $|PM|$, $|MN|$, and $|PN|$ form a geometric progression, find the value of $a$.

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