
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 $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with its right focus at $F(2,0)$, left and right vertices at $A_{1}$ and $A_{2}$ respectively, and $D$ as a point on $C$ that does not coincide with $A_{1}$ and $A_{2}$, the product of the slopes of lines $DA_{1}$ and $DA_{2}$ is $3$.  
$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ Let a point $T(m,n) (m < 1)$ not on $C$, and two lines passing through $T$ intersect the right branch of $C$ at points $A$, $B$ and $P$, $Q$ respectively. If $A$, $B$, $Q$, $P$ are concyclic, find the sum of the slopes of lines $AB$ and $PQ$.

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