
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
In a shooting match, ten clay targets are arranged in three hanging columns with four targets in column $A$, three in column $B$, and three in column $C$. A marksman is to break all the targets according to the following rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
3) The marksman has to alternate between breaking a target in column $A$ and any other column until all targets in column $A$ are broken.
In how many different orders can the ten targets be broken?

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