
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 $A= \begin{pmatrix} a & 2 \\ 1 & b \end{pmatrix}$ has an eigenvector $\overrightarrow{\alpha}= \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ corresponding to the eigenvalue 1.

(Ⅰ) Find the matrix $A$;  
(Ⅱ) If the matrix $B= \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$, find the equation of the image of the line $x+y+1=0$ under the transformation of matrix $A$ first, and then under the transformation of matrix $B$.

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