
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 the math test paper of the new college entrance examination in 2021, the scoring rules for each multiple-choice question are as follows: 5 points for all correct answers, 2 points for partially correct answers, and 0 points for any wrong answers. When solving the 11th and 12th multiple-choice questions, Xiao Ming usually adopts two strategies:<br/>Strategy $A$: To avoid mistakes, he selects only the option he is most confident about. This strategy takes about $3$ minutes per question.<br/>Strategy $B$: He selects all the options he believes are correct. This strategy takes about $6$ minutes per question.<br/>As the math exam approaches, Xiao Ming has determined the answering situations for the 11th and 12th questions under the two strategies through a large number of simulated training sessions:<br/>11th question: If strategy $A$ is adopted, the probability of selecting correctly is $0.8$, if strategy $B$ is adopted, the probability of partially correct selection is $0.5$, and the probability of all correct selections is $0.4.<br/>12th question: If strategy $A$ is adopted, the probability of selecting correctly is $0.7$, if strategy $B$ is adopted, the probability of partially correct selection is $0.6, and the probability of all correct selections is $0.3.<br/>If the total time spent on these two questions exceeds $10$ minutes, Xiao Ming will lose $2$ points on other questions due to time constraints. Assuming that Xiao Ming's answers to the two questions do not affect each other.<br/>$(1)$ If Xiao Ming decides to use strategy $B$ for the 11th question and strategy $A$ for the 12th question in this exam, and the total score for the 11th and 12th questions is $X$, find the distribution of $X$.<br/>$(2)$ Xiao Ming designed the following two plans before the exam:<br/>Plan $1$: Strategy $B$ for the 11th question and strategy $A$ for the 12th question;<br/>Plan $2$: Strategy $B$ for both the 11th and 12th questions.<br/>If you were Xiao Ming's guidance teacher, which plan would you support from the perspective of maximizing the overall score on the entire test paper? Please explain your reasoning.

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