
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
Selected Exercise $4-4$: Coordinate Systems and Parametric Equations

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

(I) Write the rectangular equation of curve $C$ and the standard form equation of line $l$.

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

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