
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
8. You have been kidnapped by a witch and are stuck in the *Terrifying Tower*, which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:

- For a floor  $i$ , put 2 boxes on floor  $i+5$ , 6 on floor  $i+4$ , 13 on floor  $i+3$ , 12 on floor  $i+2$ , 8 on floor  $i+1$ , and 1 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.
- For a floor  $i$ , put 1 box on floor  $i+4$ , put 3 boxes on floor  $i+3$ , 6 on floor  $i+2$ , 5 on floor  $i+1$ , and 3 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.

At the end, suppose the witch wishes to have exactly  $n$  boxes in the tower. Specifically, she wants them to be on the first 10 floors. Let  $T(n)$  be the number of distinct distributions of these  $n$  boxes that you can make. Find  $\displaystyle\sum_{n=1}^{15} T(n)$ .

*Proposed by Monkey_king1*


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