
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 Chinese Basketball Association (CBA) Finals, a best-of-seven series is played, where the first team to win four games is declared the champion. Team A and Team B are evenly matched in the finals, with each team having a probability of $\frac{1}{2}$ of winning each game. According to historical data, the ticket revenue for the first game is 4 million yuan, and for each subsequent game, the revenue increases by 1 million yuan.

$(1)$ Find the probability that the total ticket revenue for the finals is exactly 30 million yuan.

$(2)$ Let $X$ be the total ticket revenue for the finals. Find the mathematical expectation $E(X)$ of $X$.

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