
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
$P$ is a point on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0)\), and $F$ is a focus such that \(\triangle POF\) is an isosceles triangle (where $O$ is the origin). If there are exactly 8 such points $P$ that satisfy the condition, then the range of values for the eccentricity of the ellipse is:
\[
\qquad
\]

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