
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

$$
\left\{\begin{array}{l}
x^{2}+y^{2}=10(x \cos 4a + y \sin 4a) \\
x^{2}+y^{2}=10(x \sin a + y \cos a)
\end{array}\right.
$$

has two solutions \((x_1, y_1)\) and \((x_2, y_2)\) such that the distance between the points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is equal to 6.

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