
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 $f(x) = |2x-1| - |x+2|$.

$(1)$ Find the solution set of the inequality $f(x) > 0$;

$(2)$ If there exists $x_0 \in \mathbb{R}$ such that $f(x_0) + 2a^2 < 4a$, find the range of the real number $a$.

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