
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 hallway is covered entirely with tiles in a repeating pattern. Each corner of the hallway displays a similar arrangement of tiles. What fraction of the tiled hallway is made of darker tiles if the pattern repeats every $8 \times 8$ tiles, and in each top-left $4 \times 4$ square of these tiles there are 9 dark tiles?

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