
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
During military training at school, each student is given five bullets for target shooting. The rules are: during the shooting process, if a student hits or misses two consecutive shots, they must stop shooting; otherwise, they will use up all their bullets. Suppose that when Zhang shoots at the target, the hit rate for each bullet is $\dfrac{2}{3}$.

$(1)$ Calculate the probability that Zhang hits the target with only one of the first two shots.

$(2)$ Find the distribution and expectation of the number of bullets $X$ used by Zhang in the shooting process.

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