
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 points \( a \) and \( b \) in the plane, let \( a \oplus b \) be the unique point \( c \) such that \( a b c \) is an equilateral triangle with \( a, b, c \) in the clockwise orientation. Solve \( (x \oplus (0,0)) \oplus (1,1) = (1,-1) \) for \( x \).

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