
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
The equation $2x^3+5x-2=0$ has a unique real root $r$, and there exists a unique strictly increasing sequence of positive integers $a_n$, such that $$\frac {2}{5}= \lim_{n\rightarrow \infty }(r^{a_{1}}+r^{a_{2}}+r^{a_{3}}+\cdots +r^{a_{n}})$$ holds. Then $a_n=\_\_\_\_\_\_\_\_\_\_\_\_$．($n\in\mathbb{N}^*$)

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