
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  $x_1,x_2,y_1,y_2$  be real numbers satisfying the equations  $x^2_1+5x^2_2=10$ ,  $x_2y_1-x_1y_2=5$ , and  $x_1y_1+5x_2y_2=\sqrt{105}$ . Find the value of  $y_1^2+5y_2^2$ 

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