
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, a polar coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. It is known that the curve $C: \rho \sin^{2}\theta = 2a\cos \theta \left(a \gt 0\right)$. The line $l$ passing through the point $P(-2,-4)$ has the parametric equations $\left\{\begin{array}{l}{x=-2+t}\\{y=-4+t}\end{array}\right.$ (where $t$ is a parameter). Let $M$ and $N$ be the points where the line $l$ intersects the curve $C$. If $|PM|$, $|MN|$, and $|PN|$ form a geometric sequence, then the value of $a$ is ____.

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