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An ant in the plane travels in a spiral such that its position $(x(t), y(t))$ at time $t \geq 0$ is $\left(e^{t} \cos t, e^{t} \sin t\right)$. At time $t=1$, find the real part of $\ln (x(t)+i y(t))$.

1. $-2$
2. $1$
3. $0$
4. $-1$
5. $2$