
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 three vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ in the same plane, where $\overrightarrow{a}=(1,2)$:
(1) If $|\overrightarrow{c}|=2\sqrt{5}$ and vector $\overrightarrow{c}$ is in the opposite direction to vector $\overrightarrow{a}$, find the coordinates of $\overrightarrow{c}$.
(2) If $|\overrightarrow{b}|=\frac{\sqrt{5}}{2}$ and $(\overrightarrow{a}+2\overrightarrow{b})\cdot (2\overrightarrow{a}-\overrightarrow{b})=\frac{15}{4}$, find the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$.

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