
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
Let the function $f(x) = |x - 2a|$, $g(x) = |x + a|$, where $a \in \mathbb{R}$.
1. Let $a = 1$, if there exists $x$ such that $f(x) - g(x) \geq m$ holds, find the range of $m$.
2. If $f(x) + g(x) \geq 3$ always holds, find the range of $a$.

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