
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 ellipse $C\_1$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$ and the hyperbola $C\_2$, they intersect at point $M$ in the first quadrant. $\triangle MF\_1F\_2$ is an isosceles triangle with base $MF\_1$, and $|MF\_1|=2$. If the eccentricity $e$ of ellipse $C\_1$ is in $[\frac{3}{8}, \frac{4}{9}]$, then find the range of the eccentricity of hyperbola $C\_2$.

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