
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 $\{a\_n\}$ with a non-zero common difference, let $S\_n$ denote the sum of its first $n$ terms. If $S\_5=30$, and $a\_1$, $a\_2$, $a\_4$ form a geometric sequence:

(1) Find the general term formula for the sequence $\{a\_n\}$;
(2) Calculate the value of $\frac{1}{a\_1a\_2} + \frac{1}{a\_2a\_3} + ... + \frac{1}{a\_{20}a\_{21}}$.

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