
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 \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3.  \] For any reals  $r,s,t$ , we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|.  \] Find the minimum value of  $g(r,s,t)$ .

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