
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
Let the function $f(x) = |x - 1| + |x - 2|$.

(1) Find the solution set for the inequality $f(x) \leq 3$.

(2) If the inequality $||a+b| - |a-b|| \leq |a|f(x)$ (where $a \neq 0$, $a \in \mathbb{R}$, $b \in \mathbb{R}$) always holds, find the range of the real number $x$.

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