
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 $a \neq 0$, solve for $x$ in the equation determined by the determinant
\[
\begin{vmatrix}
x + a & x & x & x\\ 
x & x + a & x & x\\ 
x & x & x + a & x\\ 
x & x & x & x + a
\end{vmatrix} = 0.
\]
Give your answer in terms of $a$.

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