The descriptive complexity of the set of arc-connected compact subsets of the plane

We compute the exact complexity of the set of all arc-connected compact subsets of $\text{boldmath}{R}^2$, which turns out to be strictly higher than the classical $\text{boldmath}{\mathrm{\Sigma }}_1^1$ and $\text{boldmath}{\mathrm{\Pi }}_1^1$ classes of analytic and coanalytic sets, but stricly lower than the class $\text{boldmath}{\mathrm{\Pi }}_2^1$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\text{boldmath}{R}^3$.