
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 the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) where \(a > b > 0\), the eccentricity \(e \in \left[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{2}}\right]\). The line \(y = -x + 1\) intersects the ellipse at points \(M\) and \(N\), and \(O\) is the origin such that \(OM \perp ON\). Determine the range of values for \(a\).

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