
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 coordinates of the four vertices of parallelogram ABCD are A(3, 1), B(-1, 1), C(-3, -1), D(1, -1). It is transformed into a rhombus A'B'C'D' under the transformation corresponding to the matrix $M= \begin{pmatrix} k & 1 \\ 0 & 2 \end{pmatrix}$ (where $k<0$).  
(I) Find the value of $k$;  
(II) Find the inverse matrix $M^{-1}$ of the matrix $M$.

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