
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 \( S = \{1, 2, \cdots, 2016\} \). For any non-empty finite sets of real numbers \( A \) and \( B \), find the minimum value of
\[ f = |A \Delta S| + |B \Delta S| + |C \Delta S| \]
where
\[ X \Delta Y = \{a \in X \mid a \notin Y\} \cup \{a \in Y \mid a \notin X\} \]
is the symmetric difference between sets \( X \) and \( Y \), and
\[ C = \{a + b \mid a \in A, b \in B\} .\]

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