
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
There are  $n$  balls numbered from  $1$  to  $n$ , and  $2n-1$  boxes numbered from  $1$  to  $2n-1$ . For each  $i$ , ball number  $i$  can only be put in the boxes with numbers from  $1$  to  $2i-1$ . Let  $k$  be an integer from  $1$  to  $n$ . In how many ways we can choose  $k$  balls,  $k$  boxes and put these balls in the selected boxes so that each box has exactly one ball?

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