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Modular inverse is not known to be in $FNC$ or $FNL$. How about for special numbers?

Given positive integers $a,t$ such that $2^t<a<2^{t+1}$ and $\mathsf{GCD}(2^t(2^{2t}-1),a)=1$, is the problem of finding $v\in\mathbb Z$ satisfying $$(2^t\pm i)v\equiv1\bmod a$$ $$1\leq v\leq a-1$$ in $FNC$ or $FNL$ when $i\in\{-1,0,+1\}$?

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