
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
The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blank questions (each worth 8 points) and 3 problem-solving questions (worth 16, 20, and 20 points respectively), with a total score of 120 points. The second round consists of 4 problem-solving questions covering plane geometry, algebra, number theory, and combinatorics. The first two questions are worth 40 points each, and the last two questions are worth 50 points each, with a total score of 180 points. 

It is known that a certain math competition participant has a probability of $\frac{4}{5}$ of correctly answering each fill-in-the-blank question in the first round, and a probability of $\frac{3}{5}$ of correctly answering each problem-solving question in the first round. In the second round, the participant has a probability of $\frac{3}{5}$ of correctly answering each of the first two questions, and a probability of $\frac{2}{5}$ of correctly answering each of the last two questions. Assuming full marks for correct answers and 0 points for incorrect answers:
1. Let $X$ denote the participant's score in the second round. Find $P(X \geq 100)$.
2. Based on the historical competition results in the participant's province, if a participant scores 100 points or above in the first round, the probability of winning the provincial first prize is $\frac{9}{10}$, while if the score is below 100 points, the probability is $\frac{2}{5}$. Can the probability of the participant winning the provincial first prize reach $\frac{1}{2}$, and explain the reason. 

(Reference data: $(\frac{4}{5})^8 \approx 0.168$, $(\frac{4}{5})^7 \approx 0.21$, $(\frac{4}{5})^6 \approx 0.262$.)

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