Calculator Keystrokes

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 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 last three digits of the \(2014^\text{th}\) smallest ambivalent number.

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