
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
Let the equations of two lines be $x + y + a = 0$ and $x + y + b = 0$, respectively. Given that $a$ and $b$ are the two real roots of the equation $x^2 + x + c = 0$, and $0 \leq c \leq 1$, find the maximum and minimum distances between these two lines.

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