
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 hyperbola $C\_1$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with its left focus $F\_1$, it forms a tangent line with the circle $C\_2$: $x^{2} + y^{2} = a^{2}$ at point $M$. Extend $F\_1M$ to intersect the parabola $C\_3$: $y^{2} = 2px (p > 0)$ at point $N$. Hyperbola $C\_1$ and parabola $C\_3$ share a common focus. If $|MF\_1| = |MN|$, find the eccentricity of the hyperbola $C\_1$.

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