then it reduces to 

a = (7+-sqrt(47))/(7*sqrt(2)).

Given the large power in what remains, I find it hard to believe there is a strictly numerical answer, but given that we have a, and using the binomial theorem for a^x + b^x one can at least do the following:

Rewrite to a^777 + (1/a)^777, which (if I did it correctly) is:

(a + 1/a) *[ Sum k=-388 to 388 of  (-1)^k * a^(2k)]

the series, (according to Wolfram Alpha) = (1/a)^2k

so given the 2 values for a (above), the solution seems to be

(a + 1/a) * (1/a)^776 = (1/a)^777 + (1/a)^775

not sure about simplifying further....?  There could be mistakes...