
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
Given the polar coordinates system with origin as pole and positive x-axis as polar axis, the curve C is defined by $\rho\sin^2\theta=2a\cos\theta$ (where $a > 0$). A line $l$ passes through the point $P(-2, -4)$ with the parametric equations
$$
\begin{cases} 
x = -2 + t \\
y = -4 + t
\end{cases}
$$
The line $l$ intersects the curve C at points $M$ and $N$.

1. Write down the standard equation of curve C and line $l$.
2. If $|PM|$, $|MN|$, $|PN|$ form a geometric progression, find the value of $a$.

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