
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) = $\frac{1}{2}|x-a|$ (where $a \in \mathbb{R}$),

1. Solve the inequality $|x-\frac{1}{2}| + f(x) \geq 2$ when $a=3$.
2. Let the solution set of the inequality $|x-\frac{1}{2}| + f(x) \leq x$ be M. If $[\frac{1}{2}, 1] \subseteq M$, find the range of values for the real number $a$.

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