
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

\[ 
\left\{
\begin{array}{l}
x^{2}+y^{2}=26(y \sin 2a - x \cos 2a) \\
x^{2}+y^{2}=26(y \cos 3a - x \sin 3a)
\end{array}
\right.
\]

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 10.

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