
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
A certain factory packages a certain product in boxes, with each box containing $100$ items. Each box of products must be inspected before delivery to customers. If any defective items are found during inspection, they are replaced with qualified items. During inspection, 10 items are randomly selected from the box for testing, and based on the test results, a decision is made whether to inspect all the remaining products. Let the probability of each item being defective be $p\left(0 \lt p \lt 1\right)$, and assume that the whether each item is defective is independent of others. Let $f\left(p\right)$ denote the probability that exactly 3 out of the 10 items are defective. Find the value of $p$ that maximizes $f\left(p\right)$.

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