
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
Find, with proof, the minimum positive integer n with the following property: for
any coloring of the integers  $\{1, 2, . . . , n\}$  using the colors red and blue (that is, assigning the
color “red” or “blue” to each integer in the set), there exist distinct integers a, b, c between
1 and n, inclusive, all of the same color, such that  $2a + b = c.$ 

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