
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
By finding a certain sixth-degree polynomial \( x^{6} + a_{1} x^{5} + \ldots + a_{5} x + a_{6} \) with integer coefficients, one of whose roots is \( \sqrt{2} + \sqrt[3]{5} \), write the sum of its coefficients \( a_{1} + a_{2} + \ldots + a_{6} \) in the answer.

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