
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 partition of a number \( n \) is a sequence of positive integers, arranged in descending order, whose sum is \( n \). For example, \( n=4 \) has 5 partitions: \( 1+1+1+1=2+1+1=2+2=3+1=4 \). Given two different partitions of the same number, \( n=a_{1}+a_{2}+\cdots+a_{k}=b_{1}+b_{2}+\cdots+b_{l} \), where \( k \leq l \), the first partition is said to dominate the second if all of the following inequalities hold:
\[
\begin{aligned}
a_{1} & \geq b_{1} ; \\
a_{1}+a_{2} & \geq b_{1}+b_{2} ; \\
a_{1}+a_{2}+a_{3} & \geq b_{1}+b_{2}+b_{3} ; \\
& \vdots \\
a_{1}+a_{2}+\cdots+a_{k} & \geq b_{1}+b_{2}+\cdots+b_{k} .
\end{aligned}
\]

Find as many partitions of the number \( n=20 \) as possible such that none of the partitions dominates any other. Your score will be the number of partitions you find. If you make a mistake and one of your partitions does dominate another, your score is the largest \( m \) such that the first \( m \) partitions you list constitute a valid answer.

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