
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
If the set \( S = \{1, 2, 3, \cdots, 16\} \) is arbitrarily divided into \( n \) subsets, there must exist some subset that contains elements \( a, b, \) and \( c \) (which can be the same) such that \( a + b = c \). Find the maximum value of \( n \).

**Note**: If the subsets \( A_1, A_2, \cdots, A_n \) of set \( S \) satisfy the following conditions:
1. \( A_i \neq \varnothing \) for \( i = 1, 2, \cdots, n \);
2. \( A_i \cap A_j = \varnothing \);
3. \( \bigcup_{i=1}^{n} A_i = S \),

then \( A_1, A_2, \cdots, A_n \) are called a partition of set \( S \).

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