
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
Define max$\{x, y\} = \begin{cases} x, & x \geq y \\ y, & x < y \end{cases}$, and let $f(x) = \max\{a^x - a, -\log_a x\}$ ($x \in \mathbb{R}^+$, $a > 0$, $a \neq 1$). If $a = \frac{1}{4}$, then $f(2) + f\left(\frac{1}{2}\right) = \_\_\_\_\_\_$; if $a > 1$, then the solution set of the inequality $f(x) \geq 2$ is $\_\_\_\_\_\_$.

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