
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 that points $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a > 0, b > 0)$, and $P$ is a point on the right branch. Denote the distance from point $P$ to the right directrix as $d$. If $|PF\_1|$, $|PF\_2|$, and $d$ form an arithmetic sequence, then determine the range of the eccentricity of the hyperbola.

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