
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
Point \( B \) is on line segment \( \overline{AC} \) with \( AB = 15 \) and \( BC = 15 \). Point \( D \) is not on \( \overline{AC} \) such that \( AD = CD \), and \( AD \) and \( BD \) are integers. Let \( s \) be the sum of all possible perimeters of \( \triangle ACD \). Find \( s \).

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