
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
In the plane rectangular coordinate system $Oxy$, the vertices of acute angles $\alpha$ and $\beta$ coincide with the origin $O$, and the initial side coincides with the non-negative half-axis of the $x$-axis. The terminal sides intersect the unit circle $O$ at points $P$ and $Q$, respectively. It is known that the ordinate of point $P$ is $\frac{4\sqrt{3}}{7}$, and the abscissa of point $Q$ is $\frac{13}{14}$.  
$(1)$ Find the value of $\cos(\alpha - \beta)$.  
$(2)$ Let the opposite sides of the interior angles $A$, $B$, and $C$ of triangle $ABC$ be $a$, $b$, and $c$, respectively.  
Please choose one of the following two questions to answer. If multiple choices are made, scoring will be based on the first answer.  
① If $C = \alpha - \beta$ and $c = 2$, find the maximum perimeter of triangle $ABC$.  
② If $A = \alpha$, $B = \beta$, and $c = 11$, find the area of triangle $ABC.

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