
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 $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $E$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, and point $M$ is on $E$, with $MF\_1$ perpendicular to the $x$-axis and $\sin \angle MF\_2F\_1 = \frac{1}{3}$. Find the eccentricity of $E$.

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