
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 the triangle \( \triangle ABC \), \( BD = DC \). A piece of cheese is placed on side \( AC \) at the point which is one-fourth the distance from point \( C \). On \( AD \), there are three mirrors \( W_1, W_2, W_3 \) which divide \( AD \) into four equal parts. A very paranoid mouse is crawling on \( AB \) (moving from \( A \) to \( B \)), where \( AB = 400 \) meters. When the mouse, one of the mirrors, and the cheese are collinear, the mouse can see the cheese. Due to its paranoia, the mouse wants to see the cheese multiple times to ensure that the cheese has not been eaten by another mouse before it reaches it. The mouse crawls forward 80 meters in the first minute, then retreats 20 meters in the second minute, crawls forward 80 meters in the third minute, retreats 20 meters in the fourth minute, and so on. When the mouse reaches point \( B \), it will directly run along \( BC \) to eat the cheese. 

How many times in total can the mouse see the cheese while on segment \( AB \)?

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