
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 workshop has $50$ workers who need to complete the production task of $150$ products. Each product consists of a combination of $3$ type $A$ components and $1$ type $B$ component. Each worker can process $5$ type $A$ components or $3$ type $B$ components per hour. Now, these workers are divided into two groups to work simultaneously, with each group processing the same type of component. Let $x$ denote the number of workers processing type $A$ components $(x \in \mathbb{N}^*)$.
$(1)$ Express the time required to complete the processing of type $A$ components, $f(x)$, in terms of $x$;
$(2)$ To complete the entire production task in the shortest time possible, what value should $x$ take?

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