
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
Let \( t = \left( \frac{1}{2} \right)^{x} + \left( \frac{2}{3} \right)^{x} + \left( \frac{5}{6} \right)^{x} \). Find the sum of all real solutions of the equation \( (t-1)(t-2)(t-3) = 0 \) with respect to \( x \).

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