
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 quadratic function $f(x) = x^2 - 2bx + a$, it satisfies $f(x) = f(2-x)$, and the equation $f(x) - \frac{3}{4}a = 0$ has two equal real roots.  
(1) Find the expression of the function $f(x)$.  
(2) When $x \in [t, t+1]$ ($t > 0$), find the minimum value of the function $f(x)$.

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