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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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...

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system
Let \( S = \{1, 2, \ldots, n\} \), where \( n \geq 1 \). Each of the \( 2^n \) subsets of \( S \) is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set \( T \subseteq S \), we then write \( f(T) \) for the number of subsets of \( T \) that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets \( T_1 \) and \( T_2 \) of \( S \),
\[ 
f(T_1) f(T_2) = f(T_1 \cup T_2) f(T_1 \cap T_2).
\]

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