
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 $f(x) = x^2 - 2ax + 5$ where $a > 1$.
(1) If the domain and range of the function $f(x)$ are both $[1, a]$, find the value of the real number $a$;
(2) If $f(x)$ is a decreasing function on the interval $(-\infty, 2]$, and for any $x_1, x_2 \in [1, a+1]$, $|f(x_1) - f(x_2)| \leq 4$, find the range of the real number $a$.

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