
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 $a \in \mathbb{R}$, the function $f(x) = x^2 - 2ax + 5$.
(1) If the inequality $f(x) > 0$ holds for any $x \in \mathbb{R}$, find the range of the real number $a$;
(2) If $a > 1$, and the domain and range of the function $f(x)$ are both $[1, a]$, find the value of the real number $a$;
(3) The maximum value of the function $f(x)$ in the interval $[1, a+1]$ is $g(a)$, find the expression for $g(a)$.

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