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We consider the problem of allocating $m$ indivisible chores to $n$ agents with additive disvaluation (cost) functions. It is easy to show that there are picking sequences that give every agent (that uses the greedy picking strategy) a bundle of chores of disvalue at most twice her share value (maximin share, MMS, for agents of equal entitlement, and anyprice share, APS, for agents of arbitrary entitlement). Aziz, Li and Wu (2022) designed picking sequences that improve this ratio to $\frac{5}{3}$ for the case of equal entitlement. We design picking sequences that improve the ratio to~1.733 for the case of arbitrary entitlement, and to $\frac{8}{5}$ for the case of equal entitlement. (In fact, computer assisted analysis suggests that the ratio is smaller than $1.543$ in the equal entitlement case.) We also prove a lower bound of $\frac{3}{2}$ on the obtainable ratio when $n$ is sufficiently large. Additional contributions of our work include improved guarantees in the equal entitlement case when $n$ is small; introduction of the chore share as a convenient proxy to other share notions for chores; introduction of ex-ante notions of envy for risk averse agents; enhancements to our picking sequences that eliminate such envy; showing that a known allocation algorithm (not based on picking sequences) for the equal entitlement case gives each agent a bundle of disvalue at most $\frac{4n-1}{3n}$ times her APS (previously, this ratio was shown for this algorithm with respect to the easier benchmark of the MMS).