
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 positive integer  $k$  is said to be *good* if there exists a partition of  $ \{1, 2, 3,..., 20\}$  into disjoint proper subsets such that the sum of the numbers in each subset of the partition is  $k$ . How many *good* numbers are there?

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