
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 the smallest positive integer  $n$  that satisfies the following:

We can color each positive integer with one of  $n$  colors such that the equation  $w + 6x = 2y + 3z$  has no solutions in positive integers with all of  $w, x, y$  and  $z$  having the same color. (Note that  $w, x, y$  and  $z$  need not be distinct.)

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