
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 rabbit initially stands at the position  $0$ , and repeatedly jumps on the real line. In each jump, the rabbit can jump to any position corresponds to an integer but it cannot stand still. Let  $N(a)$  be the number of ways to jump with a total distance of  $2019$  and stop at the position  $a$ . Determine all integers  $a$  such that  $N(a)$  is odd.

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