
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
Let  $k<<n$  denote that  $k<n$  and  $k\mid n$ . Let  $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$  be such that, if  $n\leq 2013$  and  $k<<n$ , then  $f(k)<<f(n)$ . What’s the least possible value of  $M$ ?

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