
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
If $x$ and $y$ are real numbers and satisfy $|9y+1-x|=\sqrt{x-4}⋅\sqrt{4-x}$, find the value of $2x\sqrt{\frac{1}{x}}+\sqrt{9y}-\frac{{\sqrt{x}}}{2}+y\sqrt{\frac{1}{y}}$.

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