
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 $\overline{AC}$ with $AB = 12$ and $BC = 24$. Point $D$ not on $\overline{AC}$ such that $AD = CD$, and $AD$ and $BD$ are integers. Moreover, $\triangle BCD$ is isosceles with $BD = BC$. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$.

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