
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
An enterprise is optimizing and upgrading its production equipment. The upgraded equipment control system consists of $2k-1$ (where $k\in \mathbb{N}_{+}$) identical components. The probability of each component working properly is $p$ ($0 < p < 1$), and the components are independent of each other. When the control system has at least $k$ working components, the equipment operates normally; otherwise, it stops running. Let $p_{k}$ denote the probability of the equipment operating normally (for example, $p_{2}$ represents the probability of the equipment operating normally when the control system consists of 3 components; $p_{3}$ represents the probability when the control system consists of 5 components).

$(1)$ If the probability of each component working properly is $p=\frac{3}{4}$, and $k=2$, find the distribution and expectation of the number $X$ of working components in the control system.

$(2)$ Before the equipment upgrade, the production rate per unit time is $a$ units, with a profit of $1$ unit per product. After the upgrade, the production rate is three times the original, and high-end products are introduced with a probability of $\frac{1}{3}$ for each product. The profit for each high-end product is $2$ units. Express the profit $y$ per unit time after the upgrade in terms of $p_{k}$. Analyze whether increasing the number of components in the control system can increase profits while ensuring that the total number of components is odd.

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