
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
Determine the greatest possible value of $x$ expressed in the form $x=\frac{a+b\sqrt{c}}{d}$ provided that $\frac{7x}{9} + 1 = \frac{3}{x}$. Find the value of $\frac{acd}{b}$ given $a, b, c,$ and $d$ are integers.

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