
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
We call a set \( S \) on the real line \(\mathbb{R}\) superinvariant if for any stretching \( A \) of the set by the transformation taking \( x \) to \( A(x) = x_0 + a(x - x_0) \) there exists a translation \( B \), \( B(x) = x + b \), such that the images of \( S \) under \( A \) and \( B \) agree; i.e., for any \( x \in S \) there is a \( y \in S \) such that \( A(x) = B(y) \) and for any \( t \in S \) there is a \( u \in S \) such that \( B(t) = A(u) \). Determine all superinvariant sets.
Remark: It is assumed that \( a > 0 \).

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