
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
Let matrix $A= \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, matrix $A$ has an eigenvector $\alpha_1= \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ corresponding to the eigenvalue $\lambda_1=-1$, and an eigenvector $\alpha_2= \begin{bmatrix} 3 \\ 2 \end{bmatrix}$ corresponding to the eigenvalue $\lambda_2=4$. Find the value of $ad-bc$.

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