The sequence $(a_n)$ is defined by the following initial conditions and recurrence relations:

- Initial conditions: 
  \[ a_1 = 1, \quad a_2 = 0 \]

- Recurrence relations for $k \in \mathbb{N}$:
  \[ a_{2k+1} = a_k + a_{k+1}, \quad a_{2k+2} = 2a_{k+1} \]

Find the value of $a_m$ for $m = 2^{19} + 91$.

Let's think step by step and output the final answer within \boxed{}.
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