
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, a polar coordinate system is established. The polar equation of curve $C$ is $\rho \sin^2\theta = 2a\cos\theta (a > 0)$. The parametric equation of line $l$ passing through point $P(-2,-4)$ is $\begin{cases}x=-2+ \frac{\sqrt{2}}{2}t \\ y=-4+ \frac{\sqrt{2}}{2}t\end{cases} (t$ is the parameter$)$, and line $l$ intersects curve $C$ at points $A$ and $B$.

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

$(2)$ If $|PA|\cdot|PB|=|AB|^2$, find the value of $a$.

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