
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 $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\cos \alpha,\sin \alpha)$, let $\overrightarrow{c}= \overrightarrow{a}-t \overrightarrow{b}$ (where $t$ is a real number).
(1) When $t=1$, if $\overrightarrow{c}$ is parallel to $\overrightarrow{b}$, find the value of $2\cos ^{2}\alpha-\sin 2\alpha$.
(2) If $\alpha= \dfrac {\pi}{4}$, find the minimum value of $|\overrightarrow{c}|$ and the projection of vector $\overrightarrow{a}$ on the direction of $\overrightarrow{c}$.

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