
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 bikes to school every day, passing through five intersections. Assuming that the probability of hitting a red light at each intersection is independent, and the probability of encountering a red light at the first and last intersections is $p$, while it is $\frac{1}{2}$ for the other three intersections.
(1) If $p = \frac{2}{3}$, calculate the probability that the student encounters their first red light at the third intersection.
(2) If the probability of the student encountering at most one red light is not more than $\frac{5}{18}$, determine the range for the value of $p$.

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