
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, define point A at $(a, a)$. Let P be a moving point on the graph of the function $y = \frac{1}{x}$ ($x>0$). If the shortest distance between points P and A is $2\sqrt{2}$, then the set of all real numbers $a$ that satisfy this condition is __________.

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