
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 triangle $ABC$, if angle $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively, and it is known that $\cos 2A - 3\cos(B+C) = 1$.

(1) Determine the size of angle $A$.
(2) If the area of $\triangle ABC$, $S = 5\sqrt{3}$ and $b = 5$, find the value of $\sin B \sin C$.
(3) If $a = 1$, determine the range of possible values for the perimeter $l$ of $\triangle ABC$.

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