
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 $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=3$, $|\vec{b}|=1$, and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$,

(Ⅰ) Find $|\vec{a} +3 \vec{b}|$;

(Ⅱ) If the vector $\vec{a} +2 \vec{b}$ is perpendicular to $t \vec{a} +2 \vec{b}$, find the value of the real number $t$.

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