
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
In triangle $\triangle ABC$, the lengths of the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $b=a+1$, $c=a+2$.  
$(1)$ If $2\sin C=3\sin A$, find the area of $\triangle ABC$.  
$(2)$ Does there exist a positive integer $a$ such that $\triangle ABC$ is an obtuse triangle? If so, find the value of $a$; if not, explain why.

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