
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 the sets \( A = \left\{(x, y) \mid y = x^2 + 2bx + 1 \right\} \) and \( B = \{(x, y) \mid y = 2a(x + b)\} \), where \( a \) and \( b \) are negative constants, and \( A \cap B = \varnothing \), find the range of values of \( (a - \cos \theta)^2 + (b - \sin \theta)^2 \) for any \( \theta \in \left[0, \frac{\pi}{2}\right] \).

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