
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
Evie and Odette are playing a game. Three pebbles are placed on the number line: one at -2020, one at 2020, and one at \( n \), where \( n \) is an integer between -2020 and 2020. They take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if the sum of the three pebbles' positions is odd; Evie wins if the sum is even. For how many values of \( n \) can Evie guarantee victory if:
(a) Odette goes first;
(b) Evie goes first?

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