
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 has 100 workers who have taken on the total task of producing 1000 units of a certain product. Each unit consists of 9 type A devices and 3 type B devices. Each worker can process 1 type A device or 3 type B devices per hour. Now, the workers are divided into two groups to process type A and type B devices respectively. Let the number of workers processing type A devices be $x$, the time they need to complete the type A devices be $t_1$ hours, and the time the remaining workers need to complete the type B devices be $t_2$ hours. Let $f(x) = t_1 + t_2$.

$(1)$ Find the expression for $f(x)$ and state its domain;

$(2)$ What value of $x$ minimizes $f(x)$?

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