
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 positive number \( r \), let the set \( T = \left\{(x, y) \mid x, y \in \mathbb{R}, \text{ and } x^{2} + (y-7)^{2} \leq r^{2} \right\} \). This set \( T \) is a subset of the set \( S = \{(x, y) \mid x, y \in \mathbb{R}, \text{ and for any } \theta \in \mathbb{R}, \ \cos 2\theta + x \cos \theta + y \geq 0\} \). Determine the maximum value of \( r \).

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