
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 two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that are not collinear,

(1) If $\overrightarrow{a}$ and $\overrightarrow{b}$ have the same starting point, for what value of the real number $t$ do the endpoints $A$, $B$, and $C$ of the vectors $\overrightarrow{a}$, $t\overrightarrow{b}$, and $\frac{1}{3}(\overrightarrow{a}+\overrightarrow{b})$, respectively, lie on the same straight line?

(2) If $|\overrightarrow{a}| = |\overrightarrow{b}|$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, for what value of the real number $t$ is the value of $|\overrightarrow{a} - t\overrightarrow{b}|$ the smallest?

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