
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 bicycle travels due north at $\frac{1}{2}$ mile per minute on a straight path. Simultaneously, a circular fog bank with a radius of $40$ miles moves northeast at $\frac{1}{3}\sqrt{2}$ mile per minute. At time $t=0$, the center of the fog is $100$ miles due east of the bicycle. Determine $\frac{1}{2}(t_1+t_2)$, where $t_1$ and $t_2$ are the times when the bicycle enters and leaves the fog bank, respectively.

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