
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 ellipse $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and the hyperbola $C\_2$: $x^{2}- \frac{y^{2}}{4}=1$ share a common focus. One of the asymptotes of $C\_2$ intersects with the circle having the major axis of $C\_1$ as its diameter at points $A$ and $B$. If $C\_1$ precisely trisects the line segment $AB$, then the length of the minor axis of the ellipse $C\_1$ is _____.

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