
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 $\triangle ABC$, the lengths of the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The following conditions are given: ① $a=7$, $b=8$, $A=30^{\circ}$; ② $a=13\sqrt{3}$, $b=26$, $A=60^{\circ}$. Let $m$ and $n$ be the number of solutions for angle $B$ obtained from conditions ① and ② respectively. Let positive numbers $x$ and $y$ satisfy $mx+ny=3$. Find the minimum value of $\frac{1}{x}+\frac{2}{y}$.

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