Tyler has two calculators, both of which initially display zero. The first calculator has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke.

A positive integer $n$ is called \textit{ambivalent} if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.

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