
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 non-zero vectors $\vec{m}$ and $\vec{n}$ satisfy $4|\vec{m}|=3|\vec{n}|$, the angle between vector $\vec{m}$ and $\vec{n}$ is $\theta$, and $\cos \theta = \frac{1}{3}$. If $\vec{n} \perp (t\vec{m} + \vec{n})$, find the value of the real number $t$.

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