
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 square \( Q\left\{-\frac{1}{2} \leqslant x \leqslant \frac{1}{2} ;-\frac{1}{2} \leqslant y \leqslant \frac{1}{2}\right\} \), find an approximate solution to the equation

$$
y^{\prime}=\sin (x y)
$$

satisfying the initial condition

$$
\left.y\right|_{x=0}=0.1
$$

and estimate the error.

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