# 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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