
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 \( A(1, -1), B(4, 0), C(2, 2) \), the plane region \( D \) consists of all points \( P(x, y) \) that satisfy
\[ \overrightarrow{A P} = \lambda \overrightarrow{A B} + \mu \overrightarrow{A C} \ (1 < \lambda \leq a, 1 < \mu \leq b) \]

If the area of region \( D \) is 8, find the minimum value of \( a + b \).

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