
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 a certain game, the rules are as follows: in each game, there is a chance to earn points of $5$, $10$, or $20$, and only one type of points can be earned in each game; the probabilities of earning $5$, $10$, and $20$ points in each game are $2p$, $p$, and $1-3p$, respectively ($0<p<\frac{1}{3}$). Three games make up a round, and after each round, the total points for that round are calculated.  
$(1)$ Find the probability (expressed as an algebraic expression involving $p$) that a person's total points in one round do not exceed $25$.  
$(2)$ When the probability of a person's total points falling within the interval $(40,55)$ in one round is maximized, find the expected value of the total points in one round.

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