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

(1) If the solution set of the inequality $f(x + \frac{1}{2}) \geq 2m + 1 (m > 0)$ is $(-\infty, -2] \cup [2, +\infty)$, find the value of the real number $m$.

(2) If the inequality $f(x) \leq 2^{y} + \frac{a}{2^{y}} + |2x + 3|$ holds for any real number $x$ and $y \in \mathbb{R}$, find the minimum value of the real number $a$.

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