
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
A square piece of paper is folded in half along a line of symmetry. The resulting shape is then folded in half along a line of symmetry of the new shape. This process is repeated until \( n \) folds have been made, giving a sequence of \( n+1 \) shapes. If we do not distinguish between congruent shapes, find the number of possible sequences when:
(a) \( n = 3 \);
(b) \( n = 6 \);
(c) \( n = 9 \).
(When \( n = 1 \) there are two possible sequences.)

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