
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 $AB = 7$ and $BC = 17$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$ and both $AD$ and $BD$ are integers. Let $t$ be the sum of all possible perimeters of triangle $ACD$. Find $t$.

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