
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
Define the function $g(x) = \begin{cases} 1, & x \geq 0 \\ -1, & x < 0 \end{cases}$, $f(x) = x^2 - 2x(x-a) \cdot g(x-a)$.  
(1) If $f(2) = 0$, find the value of the real number $a$;  
(2) Solve the inequality $f(1) \leq f(0)$ for the real number $a$;  
(3) If the function $f(x)$ is monotonically increasing in the interval $[1, 2]$, find the range of values for the real number $a$.

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