
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 the vector $\overrightarrow {a} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ is an eigenvector corresponding to the eigenvalue $\lambda$ of matrix $A = \begin{pmatrix} 1 & a \\ -1 & 4 \end{pmatrix}$:  
(I) Determine the values of the real number $a$ and $\lambda$;   
(II) Find the inverse of matrix $A$.

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