
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 \( M(1,0) \), the line \( y = t x + 1 \) intersects the hyperbola \( \frac{x^{2}}{4} - \frac{y^{2}}{8} = 1 \) at two distinct points \( A \) and \( B \). Let the midpoint of the line segment \( AB \) be \( P \). Find the range of possible values for the slope of the line \( PM \).

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