
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
Given the sets \( A \), \( B \), and \( A \cup B \) have 12, 13, and 15 elements respectively, set \( C \) satisfies the following three conditions: (1) \( C \subseteq A \cup B\); (2) the intersection of \( C \) with both \( A \) and \( B \) has exactly 1 element; (3) \( C \nsubseteq B\). How many such sets \( C \) are there?

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