
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
The parabola \( y = a x^2 - b x + b \) intersects with the line \( y = a^2 x \) (where \( a > 0 \)) at points \( P \) and \( Q \), and the absolute value of the difference in the x-coordinates of \( P \) and \( Q \) is 1. Let the maximum distance from a point on the arc \( PQ \) of the parabola to the line be \( d \). (I) Express \( d \) in terms of \( a \); (II) Find the values of \( a \) and \( b \) that maximize \( d \).

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