
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 point P(x, y) satisfying the equation $x^{2} + y^{2} - 4y + 1 = 0$, determine the following:

(1) The maximum value of $x + y$?
(2) The range of values for $\frac{y + 1}{x}$?
(3) The extrema of $x^{2} - 2x + y^{2} + 1$?

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