
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 the set \( S \subseteq \{1, 2, 3, \cdots, 100\} \). It is known that for any two distinct elements \( a \) and \( b \) in \( S \), there exist a positive integer \( k \) and two distinct elements \( c \) and \( d \) in \( S \) (permitting \( c \) and \( d \) to be equal to \( a \) or \( b \)), such that \( c < d \), and \( a + b = c^{k} d \). Find the maximum number of elements in the set \( S \).

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