
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 the matrix $A= \begin{bmatrix} a & k \\ 0 & 1\\end{bmatrix}(k\neq 0)$, one of its eigenvectors is $α= \begin{bmatrix} k \\ -1\\end{bmatrix}$. The inverse matrix $A^{-1}$ transforms the point $(3,1)$ into $(1,1)$. Find the values of the real numbers $a$ and $k$.

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