
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with the right focus $F$ and left vertex $A$. If point $P$ lies on the ellipse $C$, $PF$ is perpendicular to the x-axis, and $\sin \angle PAF \lt \frac{\sqrt{10}}{10}$, determine the range of values for the eccentricity of the ellipse $C$.

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