
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 manuscript submitted to a certain magazine is first reviewed by two initial experts. If it passes the review of both initial experts, it will be accepted for publication; if it is rejected by both initial experts, it will not be accepted; if it only passes the review of one initial expert, it will then be reviewed by a third expert. If it passes the review of the third expert, it will be accepted, otherwise it will not be accepted. It is given that the probability of passing the review of each initial expert is $0.5$, and the probability of passing the review of the third expert is $0.3$. Each expert reviews independently.  
$(1)$ Find the probability that a manuscript submitted to the magazine is accepted out of $1$ submission;  
$(2)$ Let $X$ represent the number of manuscripts out of $4$ submitted to the magazine that are accepted. Find the distribution of $X$.

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