
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
Suppose $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference, and $S_{1}$, $S_{2}$, $S_{4}$ form a geometric sequence.
(1) Find the common ratio of the geometric sequence $S_{1}$, $S_{2}$, $S_{4}$;
(2) If $S_{2}=4$, find the general term formula for $\{a_{n}\}$;
(3) Under the condition of (2), let $b_{n}= \frac {3}{a_{n}a_{n+1}}$, and $T_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Find the largest positive integer $m$ such that $T_{n} > \frac {m}{20}$ holds for all $n∈N^{*}$.

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