
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

Compute the unique ordered pair \((x, y)\) of real numbers satisfying the system of equations:
$$
\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \quad \text{and} \quad \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4.
$$

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