
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 $f(x) = \max\{x^2 - ax + a, ax - a + 1\}$, where $\max\{x, y\} = \begin{cases} y, & x \leq y \\ x, & x > y \end{cases}$.  
(Ⅰ) If for any $x \in \mathbb{R}$, we always have $f(x) = x^2 - ax + a$, find the value of the real number $a$;  
(Ⅱ) If $a > 1$, find the minimum value $m(a)$ of $f(x)$.

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