
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
Let the left and right vertices of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ be $A$, $B$, and $Q$ be any point on the ellipse other than $A$, $B$. A line perpendicular to the $x$-axis passing through the right focus intersects the ellipse in the first quadrant at point $P$. Connect $AP$ and extend it to intersect the line $x = \frac{a^2}{c}$ at point $C$. If $\overrightarrow{AP} = \lambda \overrightarrow{PC} (1 \leq \lambda \leq 2)$ and $\tan \angle QAB \cdot \tan \angle QBA < \frac{2}{3}$, then the range of values for the eccentricity of the ellipse is ______.

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