
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 common difference $d=1$, and a geometric sequence $\{b_n\}$ with common ratio $q=2$, if $1$ is the geometric mean of $a_1$ and $b_1$, set vector $\overrightarrow{a}=(a_1,a_2)$, $\overrightarrow{b}=(b_1,b_2)$, and $\overrightarrow{a} \cdot \overrightarrow{b}=5$.  
$(1)$ Find the general formula for $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Let $c_n=2^{a_n}\log_2b_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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