
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
Given that the line segments $a$ and $b$ satisfy $a:b=3:2$, and $a+2b=28$.  
$(1)$ Find the values of $a$ and $b.  
$(2)$ If the line segment $x$ is the mean proportional between $a$ and $b$, find the value of $x.

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