
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
Three people, A, B, and C, are shooting at the same airplane. The probabilities of hitting the airplane for A, B, and C are $0.4$, $0.5$, and $0.8$ respectively. If only $1$ person hits the airplane, the probability of the airplane being shot down is $0.4$. If $2$ people hit the airplane, the probability of the airplane being shot down is $0.7$. If all $3$ people hit the airplane, the airplane will definitely be shot down. Find the probability of the airplane being shot down.

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