
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
Xiao Li walks from home to work at the company. The company requires employees to arrive no later than 8:00 a.m., otherwise it will be considered late. Xiao Li needs to pass through 4 intersections on the way to work, and the probability of encountering a red light at each intersection is $\frac{1}{2}$, and they are independent of each other. It is known that the average waiting time for each red light is 1 minute. If Xiao Li does not encounter any red lights, he can arrive at the company in just 10 minutes. Find:<br/>$(1)$ The latest time Xiao Li should leave home to ensure a probability of being not late higher than 90%;<br/>$(2)$ If Xiao Li leaves home at 7:48 a.m. for two consecutive days, the probability of being late on exactly one day;<br/>$(3)$ The average duration of Xiao Li's journey to work.

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