
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
Assume the quartic \( x^{4} - a x^{3} + b x^{2} - a x + d = 0 \) has four real roots \( \frac{1}{2} \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 2 \). Find the maximum possible value of \( \frac{\left(x_{1} + x_{2}\right) \left(x_{1} + x_{3}\right) x_{4}}{\left(x_{4} + x_{2}\right) \left(x_{4} + x_{3}\right) x_{1}} \) (over all valid choices of \( a, b, d \)).

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