
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 coordinate system and parametric equations) In the Cartesian coordinate system, polar coordinates are established with the origin as the pole and the positive x-axis as the polar axis, where both coordinate systems use the same unit length. Given the curve C: $psin^{2}\theta=2a\cos\theta$ ($a>0$), the parametric equation of the line ℓ passing through the point P (-2, -4) is
$$
\begin{cases}
x=-2+ \frac{\sqrt{2}}{2}t \\
y=-4+ \frac{\sqrt{2}}{2}t
\end{cases}
$$
The line ℓ intersects the curve C at points M and N. If the distances |PM|, |MN|, and |PN| form a geometric sequence, then the value of the real number a is __________.

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