
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 shopping mall uses the distribution of vouchers to stimulate consumption and has designed two lottery schemes. Scheme one: Customers throw two uniformly dense dice at once. If the product of the points is $12$, they will receive a voucher of $900. If the points are the same, they will receive a voucher of $600. In other cases, they will receive a voucher of $180. Scheme two: There are four balls in the box numbered $1$, $2$, $3$, and $4$ respectively (all are the same except for the numbers). Customers draw a ball from the box with replacement twice. If the numbers drawn on the two occasions are the same, they will receive a voucher of $600. If the sum of the numbers drawn on the two occasions is odd, they will receive a voucher of $a$ ($200\leqslant a \lt 600$). In other cases, they will receive a voucher of $100.<br/>$(1)$ If customer A randomly chooses one of the two lottery schemes, find the probability that A will receive a voucher of at least $600$; <br/>$(2)$ Customer B chooses the lottery scheme two. Let X denote the amount of the voucher B receives. If $E(X)\geqslant 400$, find the range of values for $a$.

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