
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 polar coordinate system, a line parallel to $\theta = \frac{\pi}{4}$ ($\rho \in \mathbb{R}$) is drawn from a point $$A(2\sqrt{5},\pi + \theta)$$ (where $\tan\theta=2$ and $\theta$ is an acute angle) passing outside the curve $$L: \rho\sin^2\theta=2a\cos\theta$$ ($a > 0$), intersecting the curve at points B and C respectively.  
(I) Write the standard equations of curve L and line l based on the polar coordinate system where the pole is the origin and the positive x-axis is the polar axis;  
(II) If $|AB|$, $|BC|$, and $|AC|$ form a geometric progression, determine the value of $a$.

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