
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 $\overrightarrow{a}=(3,4)$, $\overrightarrow{b}=(-2,1)$.  
$(1)$ If $\overrightarrow{c} \parallel (\overrightarrow{a}+2\overrightarrow{b})$ and $|\overrightarrow{c}|=\sqrt{37}$, find the coordinates of $\overrightarrow{c}$;  
$(2)$ 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}
user