
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 in triangle $\triangle ABC$, the sides $a$, $b$, $c$ opposite to the internal angles $A$, $B$, $C$ form a geometric sequence.  
$(1)$ If $cosB=\frac{3}{5}$ and the area of $\triangle ABC$ is $2$, find the perimeter of $\triangle ABC$.  
$(2)$ Find the range of values for $\frac{{sinA+cosAtanC}}{{sinB+cosBtanC}}$.

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