
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  $ f(n,k)$  be the number of ways of distributing  $ k$  candies to  $ n$  children so that each child receives at most  $ 2$  candies. For example  $ f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6$ . Determine the value of  $ f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012)$ .

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