$\large a_1^4+a_2^4+a_3^4+a_4^4+\ldots +a_{14}^4=49999$

Let there be integers $a_1,a_2,a_3,\ldots a_{14}$ , such that the above equation is satisfied.

How many $14-$ tuples of ${a_1,a_2,a_3\ldots,a_{14}}$ exist such that the above condition is met?

The answer is 0.

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Looking mod 16, $a^4$ gives remainder 1 or 0(That can be verified by looking at all the cases mod 16).But RHS is congruent to 15(mod 16), and the LHS is a sum of 14 0s and 1s which is impossible.