
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
Given an arithmetic sequence $\{a_n\}$ where all terms are distinct, and the sum of the first $5$ terms $S_5=20$, and $a_1$, $a_3$, $a_7$ form a geometric sequence, determine the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{{a_n}{a_{n+1}}} \right\}$.

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