
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 Cartesian coordinate system, given vectors $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(-2,6)$.<br/>$(1)$ If $\overrightarrow{c}∥(2\overrightarrow{a}+\overrightarrow{b})$ and $|\overrightarrow{c}|=3$, find the coordinates of vector $\overrightarrow{c}$;<br/>$(2)$ If the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+λ\overrightarrow{b}$ is ____, find the range of real number $\lambda$. Please choose one of the following conditions to complete the problem and solve it.<br/>① acute angle; ② obtuse angle.

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