
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
After learning the Pythagorean theorem, we all know "3, 4, 5; 5, 12, 13; 7, 24, 25; 9, 40, 41; ...". Observing these Pythagorean triples, we find that the legs of these triangles are all odd numbers, and there has been no interruption since 3.

$(1)$ According to the above pattern, please write down the next Pythagorean triple: ______;

$(2)$ If the first number is represented by the letter $n$ ($n$ is an odd number and $n\geqslant 3$), then the next two numbers can be represented by algebraic expressions containing $n$ as ______ and ______. Please explain why they form a Pythagorean triple using the knowledge you have learned.

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