
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 $A = \begin{bmatrix} 4 & 2 \\ a & 1 \end{bmatrix}$ which does not have an inverse, find:  
(1) the value of the real number $a$;  
(2) the eigenvectors of the matrix $A$.

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