
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 the function \( f(x) = x^2 + x - 2 \):
1. Find the explicit form of the function \( g(x) = \frac{|f(x)| - f(x)}{2} \).
2. If \( a > 0 \), determine the relationship between \( a \) and \( b \) such that the line \( y = ax + b \) intersects the curve \( y = g(x) \) at three distinct points. Additionally, illustrate the region corresponding to the points \( (a, b) \) in the coordinate plane.

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