
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 $f(x) = x^2 - 2ax + 1$.

(1) If for any real number $x$, $f(1+x) = f(1-x)$ holds, find the value of the real number $a$;  
(2) If $f(x)$ is monotonically increasing on the interval $[1, +\infty)$, find the range of values for the real number $a$;  
(3) When $x \in [-1, 1]$, find the maximum value of the function $f(x)$.

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