
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
Turbo the snail sits on a point on a circle with circumference  $1$ . Given an infinite sequence of positive real numbers  $c_1, c_2, c_3, \dots$ , Turbo successively crawls distances  $c_1, c_2, c_3, \dots$  around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant  $C > 0$  with the following property: for every sequence of positive real numbers  $c_1, c_2, c_3, \dots$  with  $c_i < C$  for all  $i$ , Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

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