
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
A new game is underway with 50 participants sitting in a circle, including the organizer, Sam. Sam starts with 3 calculators: one displaying 1, another displaying 0, and the last with -1. He performs the following actions: he squares the number on the 1-display calculator, presses the square button on the 0-display calculator, and on the -1-display calculator, he toggles between squaring and negating it on every alternate pass starting with squaring. Each participant performs the same operations in sequence, passing the calculators to the next person. After a full round, when all calculators return to Sam, he computes the sum of the numbers on the calculators. What is the final sum?

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