
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 the first term $a_1=1$ and common difference $d > 0$, and the sum of the first $n$ terms $S_n$  
$(1)$ If $S_1$, $S_2$, $S_4$ form a geometric sequence, find the sum of the first $n$ terms $T_n$ for the sequence $\left\{ \frac{a_n}{2^n} \right\}$;  
$(2)$ If $\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \frac{1}{a_3a_4} + \ldots + \frac{1}{a_na_{n+1}} > \frac{2015}{2016}$ holds for all $n\in\mathbb{N}^*$, find the range of values for $d$.

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