
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 rectangular coordinate system, the graph of the function \( y = \frac{1}{|x|} \) is denoted by \( \Gamma \). Let points \( P \) and \( Q \) on \( \Gamma \) satisfy the following conditions: \( P \) is in the first quadrant, \( Q \) is in the second quadrant, and the line segment \( P Q \) is tangent to the portion of \( \Gamma \) in the second quadrant at point \( Q \). Find the minimum value of \( |P Q| \).

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