
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
Let \( A = \{ t \mid 0 < t < 2\pi, t \in \mathbb{R} \} \) and \( B = \{ (x, y) \mid \left\{ \begin{array}{l} x = \sin t \\ y = 2 \sin t \cos t \end{array} \right., t \in A \} \). Also, let \( C(r) = \{ (x, y) \mid x^2 + y^2 \leq r^2, r > 0 \} \). Find the minimum value of \( r \) such that \( B \subseteq C(r) \).

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