
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
Observe the following Pythagorean triples:  
Group 1: $3 = 2 \times 1 + 1$, $4 = 2 \times 1 \times (1 + 1)$, $5 = 2 \times 1 \times (1 + 1) + 1$  
Group 2: $5 = 2 \times 2 + 1$, $12 = 2 \times 2 \times (2 + 1)$, $13 = 2 \times 2 \times (2 + 1) + 1$  
Group 3: $7 = 2 \times 3 + 1$, $24 = 2 \times 3 \times (3 + 1)$, $25 = 2 \times 3 \times (3 + 1) + 1$  
Group 4: $9 = 2 \times 4 + 1$, $40 = 2 \times 4 \times (4 + 1)$, $41 = 2 \times 4 \times (4 + 1) + 1$  
...  
Based on the patterns observed in the groups of Pythagorean triples above, the Pythagorean triples for the 7th group are     ; the Pythagorean triples for the nth group are     .

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