
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 a constant $a>0$, the expansion of $(ax^{2}+ \frac {1}{ \sqrt {x}})^{4}$ contains a term with $x^{3}$ whose coefficient is $\frac {3}{2}$. Find the value of $$\lim_{n\rightarrow \infty }(a+a^{2}+\cdots +a^{n}).$$

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