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Button Tap Sequence

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Rules

You will encounter a row of buttons, where red buttons indicate an "off" state and green buttons signify an "on" state. Pressing a button will toggle its state, but this can only occur under specific conditions:

1. The button immediately to the right must be in the "on" state, and all other buttons to its right must be in the "off" state.
2. The rightmost button can always be toggled, regardless of its state.

Example

Given the following row of buttons, only the 5th and last button can be toggled.

Resulting in this if you press the 5th button.

Resulting in this when you press the last button.

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Challenge

You will start with a row of buttons, all initially set to the "off" state, and you can press only one button at a time. The following sequence illustrates the minimum number of presses required to turn all buttons to the "on" state. Your challenge is to identify the patterns and determine the state of all buttons based on the total number of buttons k and the nth press in this sequence.

Example

Given k = 3 and n = 2, the state of the buttons is:

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Task

Your objective is to determine the state of the buttons after the nth press, given that all buttons start in the "off" state. You should use the same pressing patterns described in the "Challenge" section.

Input:

• k: The total number of buttons (positive integer).
• n: The nth press (nonnegative big integer).

Output:

• Return an array of bits, where each bit represents a button: a value of 1 indicates the "on" state, while a value of 0 indicates the "off" state.

Constraints:

• 0 ≤ N < 10^3013: N is the maximum number of steps required to turn all buttons to the "on" state using this pattern
• 1 ≤ k ≤ 10^3
• 0 ≤ n ≤ N

Puzzle Element:

• You should be able to figure the pattern and how it extends to bigger values of k and n.

Performance Considerations:

• It is important to note that naively iterating through all steps to determine the state of the buttons is to result in a timeout. Given the constraints of the problem, a more efficient approach is necessary to ensure that your solution runs within acceptable time limits.

Debugging:

• A flag is available to enable performance tests, which is set to "off" by default. To successfully pass the challenge, you need to either enable this flag or remove it from your code entirely. The flag can be found in the solution setup. Make sure to adjust it according to your testing needs to ensure that your implementation meets the performance requirements of the challenge.

Example

k = 3
n = 2

return [0, 1, 1]

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Good luck, have fun!