
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
In the Cartesian coordinate system \(xOy\), the equation of the hyperbola \(C\) is \(x^2 - y^2 = 1\). Find all real numbers \(a > 1\) that satisfy the following condition: Through the point \((a, 0)\), draw any two mutually perpendicular lines \(l_1\) and \(l_2\). If \(l_1\) intersects the hyperbola \(C\) at points \(P\) and \(Q\), and \(l_2\) intersects \(C\) at points \(R\) and \(S\), then \(|PQ| = |RS|\) always holds.

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