
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
Let \( S = \{1, 2, 4, 8, 16, 32, 64, 128, 256\} \). A subset \( P \) of \( S \) is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set \( Q \) is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets. (A set \( A \) is said to be a proper subset of a set \( B \) if \( A \) is a subset of \( B \) and \( A \neq B \).)

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