
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 that $A_{1}$ and $A_{2}$ are the left and right vertices of the hyperbola $x^{2}-y^{2}=2$, and $P$ is any point on the hyperbola except $A_{1}$ and $A_{2}$. If the distance from the origin $O$ to the lines $PA_{1}$ and $PA_{2}$ are $d_{1}$ and $d_{2}$ respectively, then determine the range of $d_{1} \cdot d_{2}$.

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