
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, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the equation of curve $C$ is given by $\rho\sin^2\theta=2a\cos\theta$ ($a > 0$). The parametric equation of the line $l$ passing through the point $P(-2,-4)$ 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 $l$ intersects curve $C$ at points $M$ and $N$.

$(1)$ Write down 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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