
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

Three spheres with radii $\sqrt{6}, 1$, and 1 touch each other pairwise. A plane $\gamma$ is passed through the line containing the centers $A$ and $B$ of the second and third spheres, such that the center $O$ of the first sphere is at a distance of 1 from this plane. Find the angle between the projections of the lines $OA$ and $OB$ on the plane $\gamma$ and compare it to $\arccos \frac{4}{5}$.

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