
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 $f(x) = |x + m| - |2x - 2m|$, where $m > 0$.

1. Find the solution set of the inequality $f(x) \geq \frac{1}{2}$ when $m = \frac{1}{2}$.
2. Find the range of real numbers $m$ such that for any real number $x$, there exists a real number $t$ that satisfies the inequality $f(x) + |t - 3| < |t + 4|$.

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