
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
Suppose that \( b \) and \( c \) are real numbers such that the equation

\[ 
x^{2} + b x + c = 0 
\]

has two different solutions \( x_{1} \) and \( x_{2} \). Suppose that

(a) The positive difference between \( x_{1} \) and \( x_{2} \) is 1;

(b) The positive difference between \( b \) and \( c \) is also 1.

Find all possible values of \( b \) and \( c \).

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