
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 that point \( P \) is a point on the hyperbola \( \frac{x^2}{8} - \frac{y^2}{4} = 1 \), with \( F_1 \) and \( F_2 \) being its left and right foci respectively, and \( O \) being the coordinate origin, determine the range of values for \( \frac{\left|PF_1\right| + \left|PF_2\right|}{|OP|} \).

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