
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
Square dancing and brisk walking have become popular fitness activities for the general public, but the noise and road occupation issues surrounding them often cause headaches and have become a hot topic of social concern. Many regions have enacted relevant policies to punish violations and regulate the conduct of square dancing, group walking, and other activities in order to guide the public in civilized fitness activities, promote a civilized, healthy, green, and environmentally friendly lifestyle for all, and issued the "Silent Square Dancing, Standardized Brisk Walking Initiative" proposal. Xiao Ming's mother actively exercises at home in response to the call, taking steps along a straight line with equal probability of moving forward and backward. It is known that she starts from point $O$, with a probability of $\frac{3}{4}$ of moving forward $1$ step and a probability of $\frac{1}{4}$ of moving backward $1$ step.
- $(1)$ Find the probability of returning to point $O$ after moving $4$ steps.
- $(2)$ If after moving $5$ steps she reaches point $Q$, let the random variable $X$ represent the number of steps between points $O$ and $Q$. Find the probability distribution and the expected value of $X$.

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