GAIC Math 2024 Problem 135

Let $n$ be a positive integer. Now, a frog starts jumping from the origin of the number line and makes $2^{n}-1$ jumps. The process satisfies the following conditions:

(1) The frog will jump to each point in the set $\left\{1,2,3, \cdots, 2^{n}-1\right\}$ exactly once, without missing any.

(2) Each time the frog jumps, it can choose a step length from the set $\left\{2^{0}, 2^{1}, 2^{2}, \cdots\right\}$, and it can jump either left or right.

Let $T$ be the reciprocal sum of the step lengths of the frog. When $n=2024$, the minimum value of $T$ is _____.