
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 $\xi$ and $\eta$ be the lifetimes of a blue and a red lightbulb, respectively. The flashlight fails when the last bulb burns out, i.e., the lifetime of the flashlight is equal to the maximum of $\xi$ and $\eta$. Obviously, $\max (\xi, \eta) \geq \eta$. Proceeding to the expectations: $\operatorname{E} \max (\xi, \eta) \geq \operatorname{E} \eta = 4$. Therefore, the expected lifetime of the flashlight is at least four years.

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