
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 $f(x)=a^{x-a}+1$ (where $a > 0$ and $a \neq 1$), it constantly passes through the fixed point $(\frac{1}{2}, 2)$,
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) If the function $g(x)=f\left(x+\frac{1}{2}\right)-1$, find the explicit formula for function $g(x)$;
(Ⅲ) Under the condition in (Ⅱ), if the function $F(x)=g(2x)-mg(x-1)$, find the minimum value $h(m)$ for $F(x)$ on the interval $[-1, 0]$.

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