
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
Elective 4-2: Matrices and Transformations
Given a matrix $ M = \left[ \begin{matrix} 1 & b \\ c & 2 \end{matrix} \right] $ has an eigenvalue $\lambda_1=4$ with a corresponding eigenvector $ \vec{e_1} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} $. Find the equation of the new curve that results from transforming the curve $5x^2+8xy+4y^2=1$ by the matrix $M$.

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