
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
The Pythagorean rule for calculating the sides of a right triangle is based on the identity:

$$
(2 n+1)^{2}+\left(2 n^{2}+2 n\right)^{2}=\left(2 n^{2}+2 n+1\right)^{2}
$$

Calculate, using this identity, the sides of right triangles for \( n=1,2,3,4,5 \).

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