
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 $f(x) = |x - a|$, where $a > 1$.

(1) When $a = 2$, find the solution set of the inequality $f(x) \geq 4 - |x - 4|$.

(2) Given that the solution set of the inequality $|f(2x + a) - 2f(x)| \leq 2$ is $\{x | 1 \leq x \leq 2\}$, find the value of $a$.

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