
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
Let the function $f(x) = x^2 + bx - 3$. For a given real number $b$, $f(x)$ has a maximum value $M(b)$ and a minimum value $m(b)$ in the interval $[b-2, b+2]$. Let $g(b) = M(b) - m(b)$.  
(1) When $b > 2$, find the expression for $g(b)$.  
(2) Find the minimum value of $g(b)$.

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