
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 sequence  $(a_n)$  as follows:  $a_0=a_1=1$  and for  $k\ge 2$ ,  $a_k=a_{k-1}+a_{k-2}+1$ .
Determine how many integers between  $1$  and  $2004$  inclusive can be expressed as  $a_m+a_n$  with  $m$  and  $n$  positive integers and  $m\not= n$ .

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