
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]


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Which quadratic equation has roots \( x_{1} \) and \( x_{2} \) that are real and satisfy the following conditions:

\[
\begin{aligned}
&x_{1} + x_{2} + x_{1} x_{2} = 2m + k \\
&(x_{1} - 1)(x_{2} - 1) = m + 1 - k
\end{aligned}
\]

If \( x_{1} - x_{2} = 1 \), what is the relationship between \( k \) and \( m \) that ensures the roots of the equation are real?

What are \( k \) and \( m \) if, in addition, \( k - m = 1 \)? Calculate \( x_{1} \) and \( x_{2} \).

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