
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 factory invested 1 million yuan at the end of 2013 to purchase a set of sewage treatment equipment. The annual operating cost of the equipment is 10,000 yuan, besides, there is a certain maintenance cost every year. The maintenance cost for the first year is 20,000 yuan, and due to the aging of the equipment, the maintenance cost increases by 20,000 yuan each subsequent year. Let the total cost of the factory using the equipment for $x$ years be $y$ (where $y$ is in ten-thousands of yuan).

(1) Express $y$ as a function of $x$ (Total cost = Purchase cost + Operating cost + Maintenance cost);
(2) Determine the optimal usage life of the equipment (i.e., the number of years that result in the lowest annual average cost).

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