
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
The first equation of the system defines a family of lines passing through the point $M(7, 1)$ and having a slope $k = \tan(a)$. The second equation defines a pair of concentric circles centered at $O(4, 4)$ with radii 1 and 2. The system will have a unique solution if and only if the line is tangent to the circle with the larger radius. There are two such lines:

The vector $\overrightarrow{M O}$ has coordinates $\{4-7, 4-1\}=\{-3, 3\}$, and thus the line $MO$ forms an angle of $3 \pi / 4$ with the x-axis. According to the Pythagorean theorem, the length of segment $MO$ is $3 \sqrt{2}$. The length of segment $OK$ (where $K$ is the point of tangency) is 2 (the radius of the larger circle). Therefore, from the right triangle $OMK$, we find the angle

$$
\varphi=\angle OMK=\arcsin \frac{2}{3 \sqrt{2}}=\arcsin \frac{\sqrt{2}}{3}
$$

Now we find the required angle

$$
a=\left(\frac{3 \pi}{4} + \pi n\right) - \varphi = \frac{3 \pi}{4} - \arcsin \frac{\sqrt{2}}{3} + \pi n
$$

Similarly, we find the value of the parameter $a$ for the second line. Since the triangles $OMK$ and $OMN$ are equal, the angle $\angle OMN$ is also

$$
\varphi = \arcsin \frac{\sqrt{2}}{3}
$$

Therefore, 

$$
a = \left(\frac{3 \pi}{4} + \pi n\right) + \varphi = \frac{3 \pi}{4} + \arcsin \frac{\sqrt{2}}{3} + \pi n
$$

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