
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
In an opaque cloth bag, there are a total of $5$ balls, all of which are identical in shape, size, and material, with some being red and some being white. Xiao Ming conducts multiple ball-picking experiments and records the data in the table below:

| Number of Picks | $10$ | $20$ | $40$ | $60$ | $100$ | $150$ | $200$ |
|-----------------|------|------|------|------|-------|-------|-------|
| Red Ball Count  | $5$  | $9$  | $18$ | $26$ | $41$  | $61$  | $81$  |
| Red Ball Frequency | $0.5$ | $0.45$ | $0.45$ | $0.433$ | $0.41$ | $0.407$ | $0.405$ |

$(1)$ Estimate the probability of randomly picking a red ball from this bag to be ________;  
$(2)$ If two balls are randomly picked from this bag, use a tree diagram or list method to calculate the probability of picking exactly one red ball and one white ball.

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