
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
A sequence of integers  $\{f(n)\}$  for  $n=0,1,2,\ldots$  is defined as follows:  $f(0)=0$  and for  $n>0$ ,  $$ \begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix} $$  Derive an explicit formula for  $f(n)$  when  $n\equiv0\pmod6$ , showing all necessary details in your derivation.


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