SHARYGIN 2019 Problem 12

Let $A_1A_2A_3$ be an acute-angled triangle inscribed into a unit circle centered at $O$ . The cevians from $A_i$ passing through $O$ meet the opposite sides at points $B_i$ $(i = 1, 2, 3)$ respectively. [list=a] [] Find the minimal possible length of the longest of three segments $B_iO$ . [] Find the maximal possible length of the shortest of three segments $B_iO$ . [/list]