
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 the function $f(x) = |x - 2a| + |x + \frac{1}{a}|$  
(1) When $a=1$, find the solution set of the inequality $f(x) > 4$;  
(2) If the inequality $f(x) \geq m^2 - m + 2\sqrt{2}$ holds for any real numbers $x$ and $a$, find the range of the real number $m$.

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