
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

$$
\left\{\begin{array}{l}
x^{2} + y^{2} = 26(y \sin a - x \cos a) \\
x^{2} + y^{2} = 26(y \cos 2a - x \sin 2a)
\end{array}\right.
$$

has two solutions \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) such that the distance between points \(P(x_{1}, y_{1})\) and \(Q(x_{2}, y_{2})\) is equal to 24.

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