
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
Three nonzero real numbers  $ a,b,c$  are said to be in harmonic progression if  $ \frac {1}{a} \plus{} \frac {1}{c} \equal{} \frac {2}{b}$ . Find all three term harmonic progressions  $ a,b,c$  of strictly increasing positive integers in which  $ a \equal{} 20$  and  $ b$  divides  $ c$ .


[17 points out of 100 for the 6 problems]

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