
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
Given any two positive real numbers $x$ and $y$, define $x \, \Diamond \, y$ as a positive real number according to some fixed rule. Assume this operation satisfies the equations $(xy) \, \Diamond \, y=x(y \, \Diamond \, y)$ and $(x \, \Diamond \, 2) \, \Diamond \, x = x \, \Diamond \, 2$ for all $x,y>0$. Also, suppose that $2 \, \Diamond \, 2=4$. Find the value of $5 \, \Diamond \, 20$.

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