
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
On a table, there is a certain quantity \( N \) of candies. Aline and Bruna agree that, alternately, each must eat at least one but no more than half of the existing quantity. The winner of the game is the one who eats the last candy. Aline always starts the game.

a) For \( N = 5 \), which of the two has the winning position? (A winning position is one in which the player can ensure a win regardless of the opponent's sequence of moves.)

b) For \( N = 20 \), which of the two has the winning position?

c) What are the values of \( N \), where \( 100 < N < 200 \), that give Bruna the winning position?

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