If \(f (x) = x^7 - p x^6 + q x^5 - r x^4 + s x^3 - t x^2 + u x - 5027 \) such that:

\(f (1) = 1\)

\(f (2) = 1\)

\(f (3) = 2\)

\(f (4) = 3\)

\(f (5) = 5\)

\(f (6) = 8\)

\(f (7) = 13\)

Given that \(p, q, r, s, t\) and \(u\) are positive rational numbers,

then \(find\) the \(sum\) of \(denominators\) of \(p, q, r, s, t\) and \(u\),

where all of them are \('false'\) fractions of numerator > denominator,

which have been simplified with GCD (numerator, denominator) = 1.

GCD: Greatest Common Divisor (for positive integers), for example GCD (24, 10) = 2. \(False\) \(fraction\) is an expression commonly used by Chinese communities.

Inspired by original question set by Dev Sharma.

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