
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 equation of a circle: $y^2 - 6y\sin\theta + x^2 - 8x\cos\theta + 7\cos^2\theta + 8 = 0$.
(1) Find the parametric equation of the trajectory of the circle's center $C$;
(2) Point $P(x, y)$ is a moving point on curve $C$ from (1), determine the range of values for $2x + y$.

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