= Solution
The continuous closed curve $q(t)=\gamma_2(t)/\gamma_1(t)$ satisfies $|q(t)-1|<1$. It lies in the right half-plane, which admits a single continuous <complex argument> in $(-\pi/2,\pi/2)$. Since $q$ closes, this argument has equal endpoint values, giving $w(q)=0$. Additivity under multiplication then yields
$$
\boxed{w(\gamma_2)=w(\gamma_1)+w(q)=w(\gamma_1).}
$$
The strict inequality ensures that the quotient stays in a region avoiding zero and admitting a single argument branch.
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