
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
A precision mechanics workshop ordered 120 steel balls with a diameter of 6.1 mm. They received 120 balls, but measurements showed that the diameters did not meet the precision requirements. The measurements revealed:

- 10 balls have a diameter of 6.01 mm
- 6 balls have a diameter of 6.11 mm

Another workshop agreed to accept the balls under the condition that they are delivered in two boxes: one for larger balls and one for smaller balls. Each box should have a label indicating the average diameter of the balls inside.

The problem is to determine the threshold diameter below which the balls will go into box $A$ (and therefore balls with a diameter greater than the threshold will go into box $B$). Also, find the numbers $a$ and $b$ to be written on the boxes. These three numbers should minimize the sum of the absolute errors, where the absolute error is the absolute difference between the diameter of the ball and the number written on the box it is placed in.

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