
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 plane vectors $\overrightarrow{a}=({1,2})$ and $\overrightarrow{b}=({-3,-2})$.<br/>$(1)$ For what value of $k$, is $k\overrightarrow{a}+\overrightarrow{b}$ perpendicular to $\overrightarrow{a}-3\overrightarrow{b}$?<br/>$(2)$ If the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+λ\overrightarrow{b}$ is acute, find the range of real number $\lambda$.

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