
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
Let  $S$  be a nonempty closed bounded convex set in the plane. Let  $K$  be a line and  $t$  a positive number. Let  $L_1$  and  $L_2$  be support lines for  $S$  parallel to  $K_1$ , and let  $ \overline {L} $  be the line parallel to  $K$  and midway between  $L_1$  and  $L_2$ . Let  $B_S(K,t)$  be the band of points whose distance from  $\overline{L}$  is at most  $ \left( \frac {t}{2} \right) w $ , where  $w$  is the distance between  $L_1$  and  $L_2$ . What is the smallest  $t$  such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all  $S$ ? ( $K$  runs over all lines in the plane.)

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