
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
Spheres with centers at points \( O_1 \) and \( O_2 \) with radii 3 and 1 respectively, touch each other. Through point \( M \), which is 3 units away from \( O_2 \), two lines are drawn, each of which is tangent to both spheres, with the points of tangency lying on the same side of point \( M \). Find the angle between the tangent lines, given that one of them forms a \( 45^\circ \) angle with the line \( O_1 O_2 \).

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