
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
Two people, A and B, form a team called "Star Team" to participate in a guessing idiom activity. In each round, A and B each guess an idiom. It is known that the probability of A guessing correctly in each round is $\frac{2}{3}$, and the probability of B guessing correctly is $p$. In each round of the activity, the guesses of A and B do not affect each other, and the results of each round do not affect each other. It is known that the probability of "Star Team" guessing one idiom correctly in the first round of the activity is $\frac{1}{2}$.  
$(1)$ Find the value of $p$;  
$(2)$ Let $X$ denote the total number of idioms guessed correctly by "Star Team" in two rounds of activity. Find the probability distribution and expectation of $X$.

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