
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 the function \( f(x) = x - \frac{2}{x} - a \ln x \) where \( a \in \mathbb{R} \) and \( a > 0 \).
1. Discuss the monotonicity of \( f(x) \).
2. If \( f(x) \) has two critical points \( x_1 \) and \( x_2 \), denote the slope of the line passing through points \( A(x_1, f(x_1)) \) and \( B(x_2, f(x_2)) \) as \( k \). Is there an \( a \) such that \( k = 2 - a \)? If it exists, find the value of \( a \); if it does not exist, explain why.

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