
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 certain factory produces a batch of precision parts, with a requirement of $\Phi 50{}_{-0.03}^{+0.04}$ ($\Phi$ represents the diameter of a circular workpiece, unit: $mm$). Five parts were randomly sampled, and the data is shown in the table below. If the measurement exceeds the requirement, it is recorded as a positive number; if it falls short, it is recorded as a negative number.

| $1$st | $2$nd | $3$rd | $4$th | $5$th |
|-------|-------|-------|-------|-------|
| $+0.031$ | $-0.037$ | $+0.018$ | $-0.021$ | $+0.042$ |

$(1)$ Which parts meet the requirement?  
$(2)$ Among the parts that meet the requirement, which one has the best quality? Explain using the concept of absolute value.

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