
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 $M= \begin{bmatrix} 1 & b \\ c & 2\end{bmatrix}$ has an eigenvalue $\lambda_1=4$ and a corresponding eigenvector $\overrightarrow {e_{1}}= \begin{bmatrix} 2 \\ 3\end{bmatrix}$.  
(1) Find the matrix $M$;  
(2) Write the inverse matrix of $M$.

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