
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 the hyperbola $\frac{x^{2}}{8} - \frac{y^{2}}{12} = 1$ with left and right foci $F_1$ and $F_2$, respectively, determine the value of the real number $t$ for a point $M$ on the right branch such that ($\overrightarrow{OM} + \overrightarrow{OF_{2}}$) $\cdot$ $\overrightarrow{F_{2}M}$ $= 0$ ($O$ is the coordinate origin), and $|\overrightarrow{F_{1}M}| = t|\overrightarrow{F_{2}M}|$.

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