
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 \( x^{2} + a x + 8 = 0 \) has two distinct roots \( x_{1} \) and \( x_{2} \); moreover,

\[ 
x_{1} - \frac{64}{17 x_{2}^{3}} = x_{2} - \frac{64}{17 x_{1}^{3}} 
\]

Find all possible values of \( a \).

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