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

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