
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 rectangular coordinate system on the plane, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is given by $ρ \sin^2 θ = a \cos θ (a > 0)$. The parameter equation of line $l$ passing through point $P(-2, -4)$ is given by $\begin{cases} x = -2 + \frac{\sqrt{2}}{2}t \\ y = -4 + \frac{\sqrt{2}}{2}t \end{cases}$ ($t$ is a parameter). Line $l$ intersects with curve $C$ at points $A$ and $B$.
(I) Write the rectangular coordinate equation for curve $C$ and the general form equation for line $l$.
(II) If $|PA| \cdot |PB| = |AB|^2$, find the value of $a$.

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