
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 $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it is known that $S_4-S_2=7a_1$, $S_5=30$.  
$(1)$ Find the general formula for $a_n$;  
$(2)$ Let $b_n= \frac {1}{S_n}$, the sum of the first $n$ terms of the sequence $\{b_n\}$, $T_n < \log_2(m^2-m)$, holds for any $n\in N^*$. Find the range of the real number $m$.

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