
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
Two segments \(AB\) and \(BC\) of lengths \(a\) and \(b\) are connected by a hinge at point \(B\); an equilateral triangle \(ACD\) is constructed on the segment \(AC\) (points \(B\) and \(D\) lie on opposite sides of the line \(AC\)). For what angle \(ABC\) will the length of segment \(BD\) be

a) the greatest;

b) the smallest?

What are the maximum and minimum possible lengths of segment \(BD\)?

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