
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
Consider the matrix $M$ defined by:
\[
M = \begin{pmatrix} x + 2a & x-a & x \\ x & x + 2a & x-a \\ x-a & x & x + 2a \end{pmatrix}.
\]
Given that $a \neq 0$, solve for $x$ when $\det(M) = 0$.

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