
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](https://artofproblemsolving.com/wiki/index.php/Equation) $2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$ has three [real](https://artofproblemsolving.com/wiki/index.php/Real) [roots](https://artofproblemsolving.com/wiki/index.php/Root). Given that their sum is $m/n$ where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) [positive integers](https://artofproblemsolving.com/wiki/index.php/Positive_integer), find $m+n.$

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