Complexity of a problem remotely related to the discrete logarithm $A=x g^x$

Let $x,g \in \mathbb{F}_p^\ast$.

Given $g$ and either $$A = x g^ x$$ or $$A = x g^{x^2-1}$$

find $x$.

What is the complexity of solving this?

Is there a reduction to the discrete logarithm in the multiplicative group?

I suspect generic exponential algorithms will work.

The solution over $\mathbb{C}$ containts Lambert W function, so a reduction might not be possible.

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