
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
The equation \(x^{2} + ax + 3 = 0\) has two distinct roots \(x_{1}\) and \(x_{2}\). Additionally, the following relation holds:

\[x_{1}^{3} - \frac{99}{2 x_{2}^{2}} = x_{2}^{3} - \frac{99}{2 x_{1}^{2}}\]

Find all possible values of \(a\).

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