
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 $\overrightarrow{m}$ and $\overrightarrow{n}$ that satisfy $|\overrightarrow{m}| = 2|\overrightarrow{n}|$ and $\cos < \overrightarrow{m}, \overrightarrow{n} > = \frac{1}{3}$, find the value of the real number $t$ if $\overrightarrow{m} \perp (t\overrightarrow{n} + \overrightarrow{m})$.

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