
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
Define a binary operation $x \, \star \, y$ for all $x, y > 0$ such that $x \, \star \, y$ simplifies to $\log_y x$. Suppose the operation $x \, \star \, y$ satisfies $(x^2 \, \star \, y) \, \star \, y = x \, \star \, y$ and $x \, \star \, (y \star y) = (x \star y) \star (x \, \star \, 1)$. Given that $1 \, \star \, 1 = 0$, find the value of $15 \, \star \, 25$.

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