
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
Milly chooses a positive integer  $n$  and then Uriel colors each integer between  $1$  and  $n$  inclusive red or blue. Then Milly chooses four numbers  $a, b, c, d$  of the same color (there may be repeated numbers). If  $a+b+c= d$  then Milly wins. Determine the smallest  $n$  Milly can choose to ensure victory, no matter how Uriel colors.

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