
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
 $ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$   satisfy the given conditions

 $ a)$   $ f(0,0)\equal{}1$  ,  $ f(0,1)\equal{}1$  , 

 $ b)$   $ \forall k \notin \left\{0,1\right\}$    $ f(0,k)\equal{}0$  and

 $ c)$   $ \forall n \geq 1$  and  $ k$  ,  $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$ 

find the sum  $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$ 

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