
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
The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ intersects with the line $x+y=2$ 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 $\frac{\sqrt{3}}{3} \leqslant e \leqslant \frac{\sqrt{6}}{3}$, find the range of the length of the major axis of the ellipse.

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