
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
Let $a > 0$ be a constant, and a moving point $M(x, y)$ ($y \neq 0$) has the product of the slopes of the lines connecting it to two fixed points $F_1(-a, 0)$ and $F_2(a, 0)$ equal to a constant value $\lambda$. If the trajectory of point M is a hyperbola with eccentricity $\sqrt{3}$, determine the value of $\lambda$.

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