
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
Point $P(x, y)$ lies on the circle $x^2 + (y - 1)^2 = 1$. In order for the inequality $x + y + c \geq 0$ to always hold, determine the range of values for the real number $c$.

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