
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 TV station is holding a live broadcast event with three levels. There are only two prizes: first prize and second prize. Contestants participating in the event must pass each level in order, and only by passing the current level can they advance to the next level. It is known that the pass rates for the first, second, and third levels are $0.6$, $0.5$, and $0.4$ respectively. Winning the first prize ($300$ yuan) requires passing all three levels, while winning the second prize ($200$ yuan) requires passing the first two levels. If a contestant wins the second prize and then wins the first prize, the prize money can be accumulated to $500$ yuan. Assuming the success or failure of each level is independent, two contestants, A and B, are participating in the event.<br/>$(1)$ Find the probability that contestant A does not win any prize in the end;<br/>$(2)$ Given that both contestant A and contestant B passed the first level, find the probability that the total prize money won by A and B in the end is $700$ yuan.

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