
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 that the circle $x^2 + y^2 = 1$ is transformed by the matrix $M = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ (where $a > 0$, $b > 0$) into the ellipse $x^2 + 4y^2 = 1$, find the eigenvalues and eigenvectors of the matrix $M$.

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