
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 little ant took a trip on a plane containing two fixed points \( A \) and \( B \). The solid line graph indicates the ant's distance from point \( A \) as a function of time along its path between moments \( t = 0 \) and \( t = 9 \); the dashed line graph provides the same information with respect to point \( B \). For example, at \( t = 7 \), the ant's distance from point \( A \) was 5, and the distance from point \( B \) was 3.

a) At what moments was the ant at the same distance from \( A \) and \( B \)?

b) What is the distance between \( A \) and \( B \)?

c) At what moments was the ant on the line that connects \( A \) and \( B \)?

d) What was the distance traveled by the ant between moments \( t = 0 \) and \( t = 9 \)?

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