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Let 
$C$ be a curve in a closed orientable surface 
$F$ of genus 
$g\geq 2$ that separates 
$F$ into subsurfaces 
$\widetilde {{F}_{i} } $ of genera 
${g}_{i} $, for 
$i= 1, 2$. We study the set of roots in 
$\mathrm{Mod} (F)$ of the Dehn twist 
${t}_{C} $ about 
$C$. All roots arise from pairs of 
${C}_{{n}_{i} } $-actions on the 
$\widetilde {{F}_{i} } $, where 
$n= \mathrm{lcm} ({n}_{1} , {n}_{2} )$ is the degree of the root, that satisfy a certain compatibility condition. The 
${C}_{{n}_{i} } $-actions are of a kind that we call nestled actions, and we classify them using tuples that we call data sets. The compatibility condition can be expressed by a simple formula, allowing a classification of all roots of 
${t}_{C} $ by compatible pairs of data sets. We use these data set pairs to classify all roots for 
$g= 2$ and 
$g= 3$. We show that there is always a root of degree at least 
$2{g}^{2} + 2g$, while 
$n\leq 4{g}^{2} + 2g$. We also give some additional applications.
