
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

...

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system
Given the hyperbola \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\) with foci \(F_{1}\) and \(F_{2}\), where \(F_{1}\) is the lower focus and \(F_{2}\) is the upper focus, and \(O\) as the origin. Point \(P\) lies on the lower branch of the hyperbola, and point \(M\) is on the directrix. The conditions are:
\[
\overrightarrow{F_{2}O} = \overrightarrow{MP}, \quad \overrightarrow{F_{1}M} = \lambda\left( \frac{\overrightarrow{F_{1}P}}{\left|\overrightarrow{F_{1}P}\right|} + \frac{\overrightarrow{F_{1}O}}{\left|\overrightarrow{F_{1}O}\right|} \right) \quad (\lambda > 0)
\]

1. Find the eccentricity \(e\) of the hyperbola.
2. If the hyperbola passes through the point \(N(\sqrt{3}, 2)\), determine the equation of the hyperbola.
3. For the hyperbola passing through \(N(\sqrt{3}, 2)\), let the endpoints of the conjugate axis be \(B_{1}\) and \(B_{2}\), where \(B_{2}\) lies on the positive \(x\)-axis. Points \(A\) and \(B\) lie on the hyperbola, and \(\overrightarrow{B_{2}A} = \mu \overrightarrow{B_{2}B}\). Determine the equation of the line \(AB\) in terms of \(\overrightarrow{B_{1}A}\) and \(\overrightarrow{B_{1}B}\).

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