
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 \(r\) such that the set \(T=\left\{(x, y) \mid x, y \in \mathbf{R}\right.\) and \(\left.x^{2}+(y-7)^{2} \leqslant r^{2}\right\}\) is a subset of the set \(S=\{(x, y) \mid x, y \in \mathbf{R}\right.\) and for any \(\theta \in \mathbf{R}\), \(\cos 2\theta + x \cos \theta + y \geqslant 0\},\) determine the maximum value of \(r\).

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