
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
Solve the equation:

\[7.74 \sqrt{\log _{5} x}+\sqrt[3]{\log _{5} x}=2.\]

Solve the system of equations (7.75-7.80):

For 7.75:
\[ \begin{cases} 2-\log _{2} y=2 \log _{2}(x+y), \\ \log _{2}(x+y)+\log _{2}\left(x^{2}-x y+y^{2}\right)=1. \end{cases} \]

For 7.76:
\[ \begin{cases} y^{5 x^{2}-51 x+10}=1, \\ x y=15. \end{cases} \]

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