
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
For any real number $m$, the line $l: y = x + m$ always intersects with the hyperbola $C: \frac{x^2}{2} - \frac{y^2}{b^2} = 1$ ($b > 0$) at common points.
(1) Find the range of the eccentricity $e$ for hyperbola $C$.
(2) If line $l$ passes through the right focus $F$ of the hyperbola $C$ and intersects the hyperbola at points $P$ and $Q$, such that $\overrightarrow{FP} = \frac{1}{5} \overrightarrow{FQ}$, find the equation of the hyperbola $C$.

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