
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 the sequence $\{a_n\}$, $a_1=1$, $a_{n+1} > a_n$, $(a_n+a_{n+1}-1)^2=4a_na_{n+1}$ $(n\in\mathbb{N}^*)$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;

$(2)$ Let $b_n=\frac{1}{\sqrt[4]{a_n}}$, $T_n=b_1+b_2+\ldots+b_n$, estimate the integer part of $T_{2017}$.

(Reference data: $1.41 < \sqrt{2} < 1.42$, $44.92 < \sqrt{2018} < 45$)

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