
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: $\overrightarrow{a},\overrightarrow{b}$ are two vectors in the same plane, where $\overrightarrow{a}=(1,2)$.  
$(1)$ If $|\overrightarrow{b}|=\frac{\sqrt{5}}{2}$ and $\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{b}$, find the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$.  
$(2)$ If $\overrightarrow{b}=(1,1)$ and the angle between $\overrightarrow{a}-3\overrightarrow{b}$ and $2\overrightarrow{a}+λ\overrightarrow{b}$ is acute, find the range of real number $\lambda$.

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