
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
An ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ intersects with a line $x+y=1$ at points $P$ and $Q$, with $OP \perp OQ$, where $O$ is the coordinate origin. The eccentricity $e$ of the ellipse satisfies $\frac{\sqrt{3}}{3} \leq e \leq \frac{\sqrt{2}}{2}$. Determine the range of values for the length of the major axis of the ellipse.

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