
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 parabola \( y = -x^2 + 2x + 8 \) intersects the x-axis at points \( B \) and \( C \), and point \( D \) bisects \( B \) and \( C \). If \( A \) is a point on the parabola above the x-axis such that \(\angle BAC\) is an acute angle, determine the range of \( AD \).

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