
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
At time  $0$ , an ant is at  $(1,0)$  and a spider is at  $(-1,0)$ .  The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the  $x$ -axis.  It so happens that the ant's horizontal speed is always half the spider's.  What will the shortest distance ever between the ant and the spider be?

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