
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 school offers a course in mold processing for students. After a period of study, they plan to hold a mold processing competition. Students Xiao Ming and Xiao Hong intend to sign up for the competition. Before the competition, Xiao Ming and Xiao Hong each underwent a one-week intensive training. The table below records the number of successful mold processing attempts by the two students each day during the intensive training. Xiao Ming forgot to record the number of successful attempts on the $7$th day, but it is known that $36\leqslant a\leqslant 60$, where $a\in Z$.

|           | Day 1 | Day 2 | Day 3 | Day 4 | Day 5 | Day 6 | Day 7 |
|-----------|-------|-------|-------|-------|-------|-------|-------|
| Index $x$ | $1$   | $2$   | $3$   | $4$   | $5$   | $6$   | $7$   |
| Xiao Ming | $16$  | $20$  | $20$  | $25$  | $30$  | $36$  | $a$   |
| Xiao Hong | $16$  | $22$  | $25$  | $26$  | $32$  | $35$  | $35$  |

$(1)$ Calculate the probability that the total number of successful attempts by Xiao Ming over these $7$ days is not less than the total number of successful attempts by Xiao Hong.  
$(2)$ Based on the number of successful attempts by Xiao Ming in the first $6$ days, find the linear regression equation of the number of successful attempts $y$ with respect to the index $x$, and estimate the value of Xiao Ming's successful attempts on the $7$th day, $a$.  
Reference formulas: In the regression equation $\hat{y}=\hat{b}x+\hat{a}$, the least squares estimates of the slope and intercept are $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}=\frac{{\sum_{i=1}^n{{x_i}}{y_i}-n\overline{x}\overline{y}}}{{\sum_{i=1}^n{x_i^2}-n{{\overline{x}}^2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.  
Reference data: $1\times 16+2\times 20+3\times 20+4\times 25+5\times 30+6\times 36=582$; $1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=91$.

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