
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
Young craftsman Xiaoqiang processes a quota of $150$ parts per day. He earns $1.2$ yuan for each part processed. If the number of parts processed does not exceed the quota, he is paid based on the actual number of parts processed. If the number of parts processed exceeds the quota, he earns an additional $0.3$ yuan for each part over the quota.  
$(1)$ Find the function relationship between Xiaoqiang's daily income $y$ (in yuan) and the number of parts processed $x$ ($x \gt 0$).  
$(2)$ Given that Xiaoqiang processed the following number of parts over $10$ days as shown in the table below:  

| Number of Parts Processed | $130$ | $140$ | $150$ | $160$ | $180$ |
|---------------------------|-------|-------|-------|-------|-------|
| Frequency (Days)          | $1$   | $3$   | $1$   | $4$   | $1$   |

① Using the frequencies of the number of parts processed over these $10$ days as the probabilities of each number of parts processed, find the probability that Xiaoqiang's daily income exceeds $180$ yuan.  
② If Xiaoqiang processes parts for one more day, and the number of parts processed $m$ combined with the original data of parts processed over $10$ days forms a new dataset, find the minimum value of $m$ if the median of the new dataset is greater than the median of the original $10$-day dataset.

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