
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
Define the function  $ g(\cdot): \mathbb{Z} \to \{0,1\}$  such that  $ g(n) \equal{} 0$  if  $ n < 0$ , and  $ g(n) \equal{} 1$  otherwise. Define the function  $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$  such that  $ f(n) \equal{} n \minus{} 1024g(n \minus{} 1024)$  for all  $ n \in \mathbb{Z}$ . Define also the sequence of integers  $ \{a_i\}_{i \in \mathbb{N}}$  such that  $ a_0 \equal{} 1$  e  $ a_{n \plus{} 1} \equal{} 2f(a_n) \plus{} \ell$ , where  $ \ell \equal{} 0$  if  $ \displaystyle \prod_{i \equal{} 0}^n{\left(2f(a_n) \plus{} 1 \minus{} a_i\right)} \equal{} 0$ , and  $ \ell \equal{} 1$  otherwise. How many distinct elements are in the set  $ S: \equal{} \{a_0,a_1,\ldots,a_{2009}\}$ ?
*(Paolo Leonetti)*

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