
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
Given a matrix $A= \begin{bmatrix} a & k \\ 0 & 1 \end{bmatrix}$ ($k \neq 0$), a eigenvector of $A$ is $\vec{a} = \begin{bmatrix} k \\ -1 \end{bmatrix}$. The transformation corresponding to the inverse of matrix $A$, $A^{-1}$, maps the point (3, 1) to (1, 1). Find $a+k$.

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