
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 student rides a bike to school every day. There are 5 traffic lights on the way from home to school. It is assumed that the events of encountering a red light at each traffic light are independent of each other, and the probability is $\frac{1}{3}$.  
$(1)$ Find the distribution of the number of red lights encountered by the student on the way, denoted as $\xi$.  
$(2)$ Find the distribution of the number of intersections, denoted as $\eta$, that the student passes before encountering a red light for the first time or reaching the destination.  
$(3)$ Find the probability that the student encounters at least one red light on the way.

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