Upstairs Downstairs

Algebra Level 4

Suppose \(a_{1}, a_{2}, a_{3}\) are the first three terms of an ascending arithmetic progression, and \(b_{1}, b_{2}, b_{3}\) are the first three terms of a geometric progression. If

(i) \(a_{1} + a_{2} + a_{3} = 126\),
(ii) \(a_{1} + b_{1} = 85\),
(iii) \(a_{2} + b_{2} = 76\), and
(iv) \(a_{3} + b_{3} = 84\),

are all fulfilled, find \(a_{3} - a_{1} + b_{1} - b_{3}\).

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