Let $T colon mathbb{N}to mathbb{N}$ denote the $3x+1$ function, where $T(n)=n/2$ if $n$ is even, $T(n)=(3n+1)/2$ if $n$ is odd. As an accelerated version of $T$, we define a emph{jump} at $n ge 1$ by $textrm{jp}(n) = T^{(ell)}(n)$, where $ell$ is the number of digits of $n$ in base 2. We present computational and heuristic evidence leading to surprising conjectures. The boldest one states that for any $n ge 2^{150}$, at most four jumps starting from $n$ are needed to fall below $n$, a strong form of the Collatz conjecture.
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