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

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