
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
At the conference "Economics of Modern Times," there was an intellectual tournament in which more than 198, but fewer than 230 scientists (with either PhD or higher degrees) participated. In a match, participants would ask each other questions within a specified time frame and record correct answers. Each participant played exactly once against every other participant. The winner of a match received one point, the loser received no points, and in the event of a draw, both participants received half a point each. At the end of the tournament, it turned out that each participant scored half of all their points in matches against PhDs. How many non-PhD scientists participated in the tournament? Provide the smallest possible number as the answer.

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