Let $$a_1, a_2, \ldots$$ and $$b_1, b_2, \ldots$$ be strictly increasing sequences of positive integers such that
(a) $$a_{n+1} = a_n + a_{n-1}$$ for $$n \ge 2$$,
(b) $$b_n = 2b_{n-1}$$ for all $$n \ge 2$$,
(c) $$a_{10} = b_{10} < 2026$$.
Find the sum of all possible values of $$a_1 + b_1$$.

Writing $$a_1 = x$$, $$a_2 = y$$ and $$b_1 = k$$, repeated use of the recurrences gives $$a_{10} = 21x + 34y$$ and $$b_{10} = 2^9 k = 512k$$, so $$21x + 34y = 512k < 2026$$ and $$k \in \{1, 2, 3\}$$. Since $$y > x$$ we get $$55x < 512k$$, and reducing modulo 34 gives $$x \equiv 26k \pmod{34}$$, which leaves only $$(x, y, k) = (18, 19, 2)$$ and $$(10, 39, 3)$$. The possible values of $$a_1 + b_1$$ are 20 and 13, whose sum is 33.