
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
(1) A competition is held in \( n \) (\(n > 1\)) rounds, and a total of \( m \) medals are awarded. In the first round, 1 medal is awarded along with \(\frac{1}{7}\) of the remaining \( m-1 \) medals. In the second round, 2 medals are awarded along with \(\frac{1}{7}\) of the remaining medals, and so on, until exactly \( n \) medals are awarded in the \( n \)-th round with no medals remaining. How many rounds are there in total in this competition? How many medals are awarded in total?

(2) A circle is divided into \( n \) different sectors (\( n \geq 2 \)), labeled as \( S_1, S_2, \cdots, S_n \). Each sector can be colored using one of three colors: red, blue, or white. The coloring is such that no two adjacent sectors can be the same color. How many different ways are there to color the circle?

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