
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 $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$ respectively. Point $P$ is on the right branch of the hyperbola, $PF\_2 \perp F\_1F\_2$, $OH \perp PF\_1$, and the foot of the perpendicular is point $H$, $OH = λOF\_1$, $λ ∈ [\frac{1}{9}, \frac{1}{2}]$.

(1) Find the equations of the asymptotes of the hyperbola when $λ = \frac{1}{3}$;

(2) Find the range of the eccentricity $e$ of the hyperbola.

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