
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 affiliated middle school has four colleges: Buqing College, Jiazhen College, Hede College, and Wangdao College, with a total of 474 students. The number of students in these four colleges is $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$ respectively. If $a_{1}$, $a_{2}$, $a_{3}$ form an arithmetic sequence with a common difference of 12, and $a_{1}$, $a_{3}$, $a_{4}$ form a geometric sequence, then the number of students in Buqing College is ______.

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