
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 points $M(x_{1}, y_{1})$ and $N(x_{2}, y_{2})$ on the ellipse $C: \frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}} = 1 (a > b > 0)$, where $x_{1} \neq x_{2}$, and point $P(\frac{a}{4}, 0)$ satisfies $|PM| = |PN|$. Determine the range of eccentricity $e$ of ellipse $C$.

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