
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
An integer container \((x, y, z)\) is a rectangular prism with positive integer side lengths \(x, y, z\), where \(x \leq y \leq z\). A stick has \(x=y=1\); a flat has \(x=1\) and \(y>1\); and a box has \(x>1\). There are 5 integer containers with volume 30: one stick \((1,1,30)\), three flats \((1,2,15), (1,3,10), (1,5,6)\) and one box \((2,3,5)\).
a. How many sticks, flats and boxes are there among the integer containers with volume 36?
b. How many flats and boxes are there among the integer containers with volume 210?
c. Suppose \(n=p_{1}^{e_{1}} \cdots p_{k}^{e_{k}}\) has \(k\) distinct prime factors \(p_{1}, p_{2}, \ldots, p_{k}\), each with integer exponent \(e_{1} \geq 1, e_{2} \geq 1, \ldots, e_{k} \geq 1\) and \(k \geq 3\). How many boxes are there among the integer containers with volume \(n\)? Express your answer in terms of \(e_{1}, e_{2}, \ldots, e_{k}\). How many boxes with volume \(n=8!\) are there?

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