
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 $f(x) = |cx + a| + |cx - b|$, $g(x) = |x - 2| + c$.  
(1) When $a = 1$, $c = 2$, $b = 3$, solve the equation $f(x) - 4 = 0$;  
(2) When $c = 1$, $b = 1$, if for any $x_1 \in \mathbb{R}$, there exists $x_2 \in \mathbb{R}$ such that $g(x_2) = f(x_1)$ holds, find the range of the real number $a$.

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