
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 $f(x) = |x-1| + |x-a|$ ($a > 1$),  
(1) If the solution set of the inequality $f(x) \geq 2$ is $\{x|x \leq \frac{1}{2} \text{ or } x \geq \frac{5}{2}\}$, find the value of $a$;  
(2) For all $x \in \mathbb{R}$, $f(x) + |x-1| \geq 1$, find the range of values for the real number $a$.

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