
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 a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with left and right foci $F\_1$ and $F\_2$ respectively.

(1) If the parabola $C\_1: y^{2} = 2px (p > 0)$ has its focus at the right focus of the hyperbola, and the line connecting the intersection points passes through point $F\_2$, find the eccentricity of the hyperbola.

(2) A line segment $F\_2M$ with slope $60^{\circ}$ passes through the right focus $F\_2$, intersecting the $y$-axis at $M$ and the hyperbola at $N$. If $\overrightarrow{MF_{2}} = 4\overrightarrow{NF_{2}}$, find the eccentricity of the hyperbola.

(3) A line with slope $30^{\circ}$ passing through the right focus intersects the right branch of the hyperbola at two points. Find the range of possible eccentricities for the hyperbola.

(4) If the eccentricity $e \in [\sqrt{2}, 2]$, let the angle $\theta$ be formed by the asymptotes of the hyperbola with the real axis as its bisector. Find the range of possible values for $\theta$.

(5) If there exist two lines $x = \pm m$ intersecting the hyperbola at points $A$, $B$, $C$, and $D$, forming a square $ABCD$, find the range of possible eccentricities for the hyperbola.

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