
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
Two ants sit at the vertex of the parabola  $y = x^2$ .  One starts walking northeast (i.e., upward along the line  $y = x$  and the other starts walking northwest (i.e., upward along the line  $y = -x$ ).  Each time they reach the parabola again, they swap directions and continue walking.  Both ants walk at the same speed.  When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares.  What is the total area of these squares?

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