
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
In a plane, three vectors are given as $\overrightarrow{a}=(3,2)$, $\overrightarrow{b}=(-1,2)$, $\overrightarrow{c}=(4,1)$. Solve the following problems:<br/>$(Ⅰ)$ If the angle between the vectors $\overrightarrow{a}+\overrightarrow{b}$ and $m\overrightarrow{c}+\overrightarrow{b}$ is acute, find the range of the real number $m$;<br/>$(Ⅱ)$ If $(\overrightarrow{a}+k\overrightarrow{c})⊥(2\overrightarrow{b}-\overrightarrow{a})$, find the value of the real number $k$.

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