
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 that $$\overrightarrow{\alpha} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$$ is an eigenvector of the matrix $A = \begin{bmatrix} 1 & a \\ -1 & 4 \end{bmatrix}$ corresponding to the eigenvalue $\lambda$, find the real number $a$, the value of $\lambda$, and $A^2$.

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