
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
(1) Given that $a$, $b$, $x$, and $y$ are positive numbers satisfying $a+b=10$, $\frac{a}{x} + \frac{b}{y} = 1$, and that the minimum value of $x+y$ is 18, find the values of $a$ and $b$.
(2) If the inequality $x+2\sqrt{2xy} \leq a(x+y)$ is always true for all positive numbers $x$ and $y$, find the minimum value of the positive number $a$.

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