
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
To understand the relationship between the incidence of a disease in City $J$ and age, the following data is obtained from the Disease Control Center in City $J$:

| Age Group (years) | $[20,30)$ | $[30,40)$ | $[40,50)$ | $[50,60)$ | $[60,70)$ |
|-------------------|-----------|-----------|-----------|-----------|-----------|
| Incidence Rate (‰) | $0.09$ | $0.18$ | $0.30$ | $0.40$ | $0.53$ |

$(1)$ If the midpoint data of each interval is denoted as $x_{i}$ and the corresponding incidence rate is denoted as $y_{i}$ $(i=1,2,3,4,5)$, an empirical regression equation of the incidence rate $y$ (‰) on age $x$ (years) can be established as $\hat{y}=\hat{b}x+\hat{a}$. Find $\hat{a}$.

Given: $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}},\sum_{i=1}^5{x_i^2}=11125,\sum_{i=1}^5{{x_i}}{y_i}=78.5$

$(2)$ Medical research shows that there may be errors in test results. There is a resident in City $J$ aged between $50$ and $60$. Let $A$ represent the event "the test result of the resident is positive," $B$ represent the event "the resident has a certain disease." Given $P\left(A|B\right)=0.99$, $P({\overline{A}|\overline{B}})=0.999$, find $P\left(B|A\right)$ (round the result to $0.001$).

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