
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
In the set $\{(x, y) \mid x, y \in \mathbf{R}\}$, define the operation * as follows:
\[
\left(x_{1}, y_{1}\right) * \left(x_{2}, y_{2}\right) = \left(x_{1} x_{2}, x_{1} y_{2} + x_{2} y_{1}\right)
\]
If for any $(x, y)$, it is true that $(x, y) * (a, b) = (x, y)$, then what is $(a, b)$?

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