
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
Consider the operation $*$ that takes a pair of integers and returns an integer according to the rule
$$
a * b = a \times (b + 1).
$$

(a) For each positive integer $n$, determine all permutations $a_{1}, a_{2}, \ldots, a_{n}$ of the set $\{1, 2, \ldots, n\}$ that maximize the value of
$$
\left(\cdots\left(\left(a_{1} * a_{2}\right) * a_{3}\right) * \cdots * a_{n-1}\right) * a_{n}.
$$

(b) For each positive integer $n$, determine all permutations $b_{1}, b_{2}, \ldots, b_{n}$ of the set $\{1, 2, \ldots, n\}$ that maximize the value of
$$
b_{1} *\left(b_{2} *\left(b_{3} * \cdots *\left(b_{n-1} * b_{n}\right) \cdots \right)\right).
$$

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