
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
Let the hyperbola $\Gamma$: $\frac{x^2}{t^2} - y^2 = 1 (t > 0)$, where point $F_{1}$ is the left focus of $\Gamma$ and point $O$ is the origin.  
$(1)$ If the eccentricity of $\Gamma$ is $\frac{\sqrt{10}}{3}$, find the focal length of the hyperbola $\Gamma$;  
$(2)$ A line passing through point $F_{1}$ with a normal vector $\overrightarrow{n} = (t, -1)$ intersects a asymptote of $\Gamma$ at point $M$. If ${S_{\triangle MOF_1}} = \frac{1}{2}$, find the equation of the hyperbola $\Gamma$;  
$(3)$ If $t = \sqrt{2}$, a line $l: kx - y + m = 0 (k > 0, m \in \mathbb{R})$ intersects $\Gamma$ at points $P$ and $Q$, where $|\overrightarrow{OP} + \overrightarrow{OQ}| = 4$. Find the range of values for the slope $k$ of the line $l$.

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