
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 minimum positive integer  $k$  such that there exists a function  $f$  from the set  $\Bbb{Z}$  of all integers to  $\{1, 2, \ldots k\}$  with the property that  $f(x) \neq f(y)$  whenever  $|x-y| \in \{5, 7, 12\}$ .

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