
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
If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-collinear, non-zero vectors,  
(1) If $\overrightarrow{a}$ and $\overrightarrow{b}$ have the same starting point, for what value of the real number $t$ are the endpoints A, B, C of the vectors $\overrightarrow{a}$, $t\overrightarrow{b}$, and $\frac{1}{3}(\overrightarrow{a}+ \overrightarrow{b})$ collinear?  
(2) If $|\overrightarrow{a}|=|\overrightarrow{b}|$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is 60°, for what value of the real number $t$ is the value of $|\overrightarrow{a}-t\overrightarrow{b}|$ minimized?

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