Cyclically 91

Find the sum of integers \(c\) for all triples of integers \((a,b,c),\) \(a\leq b \leq c,\) that satisfy the system of equations \[\begin{cases} a^2-bc=91\\ b^2-ac=91\\ c^2-ab=91 \end{cases}\]

Details and assumptions

If a number \(c\) appears in several different triples \((a,b,c),\) it must be counted with multiplicity.

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