
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
Let $ABCD$ be a cyclic quadrilateral where the side lengths are distinct integers less than or equal to 13, and satisfy $AC \cdot BD = AB \cdot CD$. Find the largest possible value of $BD$.
A) $\sqrt{162}$
B) $\sqrt{172}$
C) $\sqrt{178}$
D) $\sqrt{184}$
E) $\sqrt{196}$

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