
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a bug that starts at the origin, moves 1 unit to the right, then 1 unit upward, and continues with a pattern of halving the step length after every two moves, making a $90^\circ$ turn after every second segment traveled, calculate the point where the bug's path will bring it closest after infinite moves.

Present the answer in LaTex format: \boxed{Your answer}
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