
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 the Cartesian coordinate system $xOy$, let point $A$ be a common point of curve $C_1: y=ax^3+1$ ($a>0$) and curve $C_2: x^2+y^2= \frac {5}{2}$. If the tangents to $C_1$ and $C_2$ at point $A$ are perpendicular to each other, then the value of the real number $a$ is __________.

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