
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  $N$  be a positive integer. Two persons play the following game. The first player writes a list of positive integers not greater than  $25$ , not necessarily different, such that their sum is at least  $200$ . The second player wins if he can select some of these numbers so that their sum  $S$  satisfies the condition  $200-N\le S\le 200+N$ . What is the smallest value of  $N$  for which the second player has a winning strategy?

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