
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 quadratic function $f(x) = x^2 + bx + c$, it holds that $f(x) = f(2-x)$ for all $x \in \mathbb{R}$, and $3$ is a root of $f(x)$.  
(Ⅰ) Find the expression for the function $f(x)$.  
(Ⅱ) Let $g(x) = f(a^x)$ ($a > 1$), if the maximum value of $g(x)$ on the interval $[-1, 1]$ is equal to $5$, find the value of the real number $a$.

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