
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
Place " $b$ "-shaped polyominoes on an $8 \times 8$ grid as shown in the diagram. The polyominoes must align with the grid lines. After covering, the number of polyominoes in all horizontal and vertical grid lines must be equal. How many such " $b$ "-shaped polyominoes can be placed on the grid overall? (Note: The placed polyominoes can be rotated but not flipped).

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