
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
Using $f\left(n_{1}, n_{2}, \cdots, n_{r}\right)$ to represent the number of permutations formed from $n_{1}$ elements of $a_{1}$, $n_{2}$ elements of $a_{2}$, ..., $n_{r}$ elements of $a_{r}$ where no two adjacent elements are the same (called alternating permutations), give the formula for $f\left(n_{1}, n_{2}, \cdots, n_{r}\right)$.

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