
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 $f(x) = |x - 2a| + |x - 3a|$.
1. If the minimum value of $f(x)$ is $2$, find the value of $a$.
2. If for every $x \in \mathbb{R}$, there exists an $a \in [-2, 2]$ such that the inequality $m^2 - |m| - f(x) < 0$ holds, determine the range of the real number $m$.

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