
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
Jiangsu implements the "new college entrance examination plan: 3+1+2" model, where the unified examination subjects are as follows: "3" refers to Chinese, mathematics, and foreign languages, regardless of arts or sciences. Students, based on the requirements of universities and their own strengths and interests, first choose one subject from physics and history for "1", and then choose two subjects from political science, geography, chemistry, and biology for "2". At a certain school, the proportion of students choosing physics is $\frac{3}{4}$ of the total students; and given that physics is chosen, the probability of choosing geography is $\frac{2}{3}$; and given that history is chosen, the probability of choosing geography is $\frac{4}{5}$.  
(1) Find the probability of students finally choosing geography at this school;  
(2) Let the number of students choosing geography at this school be a random variable X.  
① Find the probability of X=2;  
② Find the probability distribution table of X and its mathematical expectation.

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