
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
Find all values of the parameter \( a \) for which the system of equations

$$
\left\{\begin{array}{l}
x^{2}+y^{2}=10(x \cos a+y \sin a) \\
x^{2}+y^{2}=10(x \sin 3a+y \cos 3a)
\end{array}\right.
$$

has two solutions \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) such that the distance between the points \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) is equal to 8.

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