
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 points A $(x, y)$, B $(1, 1)$, and C $(5, 2)$, if there is a linear programming problem where the feasible region is the boundary of $\triangle ABC$ and its interior, and the linear objective function is $z=ax+by$. The minimum value of $z$ at point B is 3, and the maximum value of $z$ at point C is 12. Find the range of $ax+by$.

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