
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. The side lengths of $ABCD$ are distinct integers less than $20$ such that $AB\cdot BC=CD\cdot DA$. What is the largest possible value of $BD$?
A) $\sqrt{380}$
B) $\sqrt{392}$
C) $\sqrt{405}$
D) $\sqrt{420}$
E) $\sqrt{450}$

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