
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 \( R \) be the triangular region on the plane with vertices at points \( A(4,1), B(-1,-6), C(-3,2) \). Find the maximum and minimum values of the function \( 4x - 3y \) while \((x, y)\) varies within this triangular region (including the interior and boundary of the triangle). Provide a proof for your findings.

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