
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
(1) Given the matrix $M = \begin{bmatrix} 2 & a \\ b & 1 \end{bmatrix}$, where $a$ and $b$ are real numbers, and point $A(3, -1)$ is transformed into point $B(3, 5)$ under the action of matrix $M$, find the eigenvalues of matrix $M$.
(2) In polar coordinates, suppose the line $\theta = \frac{\pi}{3}$ intersects with the curve $\rho^2 - 10\rho\cos\theta + 4 = 0$ at points $A$ and $B$. Find the polar coordinates of the midpoint of segment $AB$.

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