
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
Develop necessary and sufficient conditions that the equation \[ \begin{vmatrix} 0 & a_1 - x & a_2 - x  -a_1 - x & 0 & a_3 - x  -a_2 - x & -a_3 - x & 0\end{vmatrix} = 0 \qquad (a_i \neq 0) \] shall have a multiple root.


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