
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 "Guess the Song Title" game, participants guess the song title based on the ringtone of the main melody of the song. A guest participates in the game show where the production team has prepared ringtone melodies of songs from two groups, $A$ and $B$. Two ringtone melodies are randomly played from each of the $A$ and $B$ groups. The guest guesses the song title based on the ringtone melodies. It is known that the probability of the guest correctly guessing the title of each song in group $A$ is $\frac{2}{3}$, and in group $B$ is $\frac{1}{2}$. The probabilities of correctly guessing the titles of each song are independent of each other.  
$(1)$ Find the probability that the guest guesses at least $2$ song titles correctly.  
$(2)$ If the guest earns $1$ point for correctly guessing a song title from group $A$ and $2$ points for correctly guessing a song title from group $B$, while earning $0$ points for a wrong guess, let $X$ denote the total score accumulated by the guest. Find the probability distribution and expectation of $X$.

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