
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 that $F$ is the focus of the parabola $y^2 = ax$ ($a > 0$). The coordinates of point $M$ are $(4, 0)$. A line passing through point $F$ with slope $k_1$ intersects the parabola at points $A$ and $B$. Extending $AM$ and $BM$ to intersect the parabola again at points $C$ and $D$, respectively, let the slope of line $CD$ be $k_2$, and $k_1 = \sqrt{2}k_2$. Find the value of $a$.

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