
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 common difference of $2$, and a geometric sequence $\{b_n\}$ with a common ratio of $2$, and $a_nb_n=n\cdot 2^n$.  
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Let $c_n= \frac {1}{a_n\cdot \log _{2}b_{n+3}}$, and denote the sum of the first $n$ terms of the sequence $\{c_n\}$ as $T_n$, compare the size of $T_n$ and $\frac {3}{8}$.

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