
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 \( A = \{t \mid 0 < t < 2\pi, t \in \mathbf{R} \}, \) and 

\[ B = \left\{(x, y) \left\lvert\,
\begin{array}{l}
x = \sin t \\
y = 2 \sin t \cos t
\end{array}, t \in A\right\}\right., \]

and suppose that 

\[ C(r) = \{(x, y) \mid x^2 + y^2 \leqslant r^2, r > 0\}. \]

Then the smallest value of \(r\) such that \( B \subseteq C(r) \) is $\qquad$ .

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