
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 $\mathbf{m}$ and $\mathbf{n}$ satisfy $4|\mathbf{m}|=3|\mathbf{n}|$ and $\cos \langle \mathbf{m},\mathbf{n} \rangle =\frac{1}{3}$. If $\mathbf{n} \perp (t\mathbf{m}+\mathbf{n})$, then the value of the real number $t$ is \_\_\_\_\_\_\_\_.

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