
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
For each permutation $\tau = (x_1, x_2, \cdots, x_{10})$ of the numbers $1,2, \cdots, 10$, define
$$
S(\tau) = \sum_{k=1}^{10} \left| 2 x_k - 3 x_{k+1} \right| \text{, }
$$

with the convention $x_{11} = x_1$. Determine:
(1) The maximum and minimum values of $S(\tau)$;
(2) The number of permutations $\tau$ that achieve the maximum value of $S(\tau)$;
(3) The number of permutations $\tau$ that achieve the minimum value of $S(\tau)$.

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