
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
Let \( x_{1}, y_{1}, x_{2}, y_{2} \) be real numbers satisfying the equations \( x_{1}^{2}+5 x_{2}^{2}=10 \), \( x_{2} y_{1}-x_{1} y_{2}=5 \) and \( x_{1} y_{1}+5 x_{2} y_{2}=\sqrt{105} \). Find the value of \( y_{1}^{2}+5 y_{2}^{2} \).

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