
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
Two squids are forced to participate in a game. Before it begins, they will be informed of all the rules, and can discuss their strategies freely. Then, they will be locked in separate rooms, and be given distinct positive integers no larger than  $2023$  as their IDs respectively. The two squids then take turns alternatively; on one's turn, the squid chooses one of the following:
 
    1.  announce a positive integer, which will be heard by the other squid;
    2.  declare which squid has the larger ID. If correct, they win and are released together; otherwise, they lose and are fried together.

Find the smallest positive integer  $N$  so that, no matter what IDs the squids have been given, they can always win in a finite number of turns, and the sum of the numbers announced during the game is no larger than  $N$ .


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