
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
Define the operation $$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$. Let the function $$f(x) = \begin{vmatrix} x - 1 & 2 \\ -x & x + 3 \end{vmatrix}$$ have a graph with vertex at the point $(m, n)$. If $k$, $m$, $n$, and $r$ form an arithmetic sequence, then find the value of $k + r$.

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