
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 $a > b > 0$, the eccentricity of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ is $e_{1}$, and the eccentricity of the hyperbola $\frac{{x}^{2}}{{b}^{2}}-\frac{{y}^{2}}{{a}^{2}-2{b}^{2}}=1$ is $e_{2}$. If $e_{1}e_{2} \lt 1$, then the range of $\frac{a}{b}$ is ______.

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