
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 two positive real numbers $a$ and $b$, define $a \, \Diamond \, b$ as a positive real number dependent on $a$ and $b$ with the rule satisfying the equations:
1. $(a^2b) \, \Diamond \, b = a(b \, \Diamond \, b)$
2. $(a \, \Diamond \, 1) \, \Diamond \, a = a^2 \, \Diamond \, 1$
Given that $1 \, \Diamond \, 1 = 1$, find the value of $3 \, \Diamond \, 5$.

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