
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
Consider an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and a line $x + y = 1$. They intersect at points $P$ and $Q$, and $OP \perp OQ$, where $O$ is the origin.
(1) Find the value of $\frac{1}{a^2} + \frac{1}{b^2}$;
(2) If the eccentricity $e$ of the ellipse satisfies $\frac{\sqrt{3}}{3} \leq e \leq \frac{\sqrt{2}}{2}$, find the range of values for the length of the major axis.

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