
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 ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passing through the point $P(2, \sqrt{2})$, with an eccentricity of $e = \frac{\sqrt{2}}{2}$, and the line $l$ with the equation $x = 4$.

1. Find the equation of the ellipse $C$;
2. A line passing through the right focus of the ellipse (not through point $P$) intersects the ellipse at two points $A$ and $B$. The line $AB$ intersects line $l$ at point $M$. If the slopes of $PA$, $PB$, and $PM$ are denoted by $k_1$, $k_2$, and $k_3$ respectively, determine whether $(k_1 + k_2 - 2k_3)$ is a constant value. If it is, find this value; otherwise, explain the reason.

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