
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 ellipse $C\_1$ and hyperbola $C\_2$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ share common foci $F\_1$, $F\_2$, they intersect at point $M$ in the first quadrant, and $\overrightarrow{F\_1 M} \cdot \overrightarrow{F\_2 M} = 0$. If the eccentricity of ellipse $C\_1$ ($e\_1$) is within the interval $[\frac{3}{4}, \frac{3\sqrt{10}}{10}]$, then the range of the eccentricity of hyperbola $C\_2$ ($e\_2$) is _______.

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