
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
Consider three collinear points \(B\), \(C\), and \(D\) such that \(C\) is between \(B\) and \(D\). Let \(A\) be a point not on the line \(BD\) such that \(AB = AC = CD\).

(a) If \(\angle BAC = 36^\circ\), then verify that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]

(b) Now, suppose that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]

Verify that \(\angle BAC = 36^\circ\).

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