
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 a batch of a certain type of light bulbs, a random sample of 200 bulbs is taken, and their lifespans are tracked and recorded in a frequency distribution table as shown in Table 1. The light bulbs are classified into three grades according to their lifespans: high-quality (lifespan ≥ 500 days), standard (300 days ≤ lifespan < 500 days), and inferior (lifespan < 300 days).

(1) Based on the data in Table 1, determine the values of $a$ and $b$;
(2) A person randomly purchases $n$ $(n \in \mathbf{Z}_{+})$ bulbs from the sample. If the $n$ bulbs' grade distribution matches the stratified sampling result of the three grades, find the minimum value of $n$;
(3) A person randomly purchases 3 bulbs from this batch for use. Using the above frequency as probabilities, let $X$ represent the number of inferior bulbs among the 3 purchased. Find the distribution and mathematical expectation of $X$.

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