
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
Noémie and Tristan are shepherds. Each of them has a number of sheep that is a perfect square, meaning an integer that can be written in the form \( n^{2} = n \times n \) with \( n \) being a non-negative integer. Let \( a^{2} \) be the number of sheep Noémie has and \( b^{2} \) be the number of sheep Tristan has. After a quick count of the total number of sheep, they conclude that \( 97 \leq a^{2} + b^{2} \leq 108 \). Determine the number of sheep each shepherd has given that:

1. Noémie has strictly more sheep than Tristan;
2. Each of them has at least 2 sheep;
3. The total number of sheep \( a^{2} + b^{2} \) is odd.

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