
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
A set is called "simple" if it does not contain elements \(x, y, z\) such that \(x + y = z\). Given the set \(\{1,2, \cdots, 2n+1\}\), find the maximum number of elements that a simple subset can contain.

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