
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 $A = \begin{bmatrix} 1 & -1 \\ a & 1 \end{bmatrix}$, if point $P(1,1)$ is transformed by matrix $A$ to point $P'(0,-8)$,
(1) Find the value of the real number $a$;
(2) Calculate the eigenvalues of matrix $A$.

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