
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
If a second-order matrix $M$ satisfies: $M \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 5 & 8 \\ 4 & 6 \end{pmatrix}$.  
(I) Find the second-order matrix $M$;  
(II) If the curve $C: x^2+2xy+2y^2=1$ is transformed by the matrix $M$ to get the curve $C'$, find the equation of the curve $C'$.

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