
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 new operation $\circ$ for any two positive real numbers $x$ and $y$, such that $x \circ y$ is defined and satisfies the equations $(xy) \circ y = x (y \circ y)$ and $(x \circ 1) \circ x = x \circ 1$ for all $x, y > 0$. Additionally, it is given that $1 \circ 1 = 2$. Determine the value of $5 \circ 7$.

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