
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 a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. One of its asymptotes is $x+\sqrt{2}y=0$. Point $M$ lies on the hyperbola, and $MF\_1 \perp x$-axis. If $F\_2$ is also a focus of the parabola $y^{2}=12x$, find the distance from $F\_1$ to line $F\_2M$.

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