
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 that the sequence $\{a_{n}\}$ is a geometric sequence with all positive integers, $a_{1}=3$, and $a_{3}$ is the arithmetic mean of $a_{2}$ and $\frac{3}{4}a_{4}$, the sequence $\{b_{n}\}$ satisfies $b_{1}=1$, $b_{n+1}=2b_{n}+1$.<br/>$(Ⅰ)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(Ⅱ)$ If $k\cdot \frac{{b_n+5}}{2}-a_n\geqslant 8n+2k-24$ holds for all $n\in N^{*}$, find the range of real number $k$.

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