
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
Let \( O \) be the pole in polar coordinates. The hyperbola \(\rho=\frac{a^{2}}{5-6 \sqrt{2} \cos \theta}\) intersects the line \(\rho \cos \left(\theta-\frac{\pi}{4}\right) = a\) at points \( A \) and \( B \). Determine the range of values for \( a \) such that the angle \(\angle A O B \leq \frac{2 \pi}{3}\), considering that \(\rho\) can take negative values.

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