
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
Boris enjoys creating shapes by dropping checkers into a Connect Four set. The number of distinct shapes possible, accounting for horizontal flips about the vertical axis of symmetry, is given by \(9(1+2+\cdots+n)\). If the total number of shapes possible is expressed as this formula, find \(n\). Note that the Connect Four board consists of seven columns and eight rows, and when a checker is dropped into a column, it falls to the lowest available position in that column. Additionally, two shapes that are mirror images of each other are considered the same shape.

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