
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 an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F_1$ and $F_2$, respectively. Let $P$ be a point on the ellipse such that $|PF_1| = λ|PF_2| (\frac{1}{2} \leqslant λ \leqslant 2)$ and $\angle F_1 P F_2 = \frac{\pi}{2}$. Determine the range of the eccentricity of the ellipse.

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