
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 quartic polynomial equation \[x^4 + ax^3 + bx^2 + cx + d = 0,\] where $a$, $b$, $c$, and $d$ are rational numbers. It is known that $3-\sqrt{5}$ and $3+\sqrt{5}$ are two of its roots. If one other root is an integer and the sum of the roots (based on Vieta's formulas) for $x^4 + ax^3 + bx^2 + cx + d = 0$ should be zero, what is the integer root?

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