
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
In the convex quadrilateral \(ABCD\), the lengths of the sides are given as \( |AB|=a \), \( |AD|=b \), \( |BC|=p-a \), \( |DC|=p-b \). Let \(O\) be the point of intersection of the diagonals. Denote by \(\alpha\) the angle \(\widehat{BAC}\). To what does the length \( AO \) approach as \(\alpha\) approaches zero?

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