
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 \( S = \{-3, -2, 1, 2, 3, 4\} \). Two distinct numbers \( a \) and \( b \) are randomly chosen from \( S \). Let \( g(a, b) \) be the minimum value of the function \( f(x) = x^2 - (a+b)x + ab \) with respect to the variable \( x \). Find the maximum value of \( g(a, b) \).

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