
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 {a_{n}} is a geometric sequence, and the function f(x)=\frac{{x}^{3}}{3}-\frac{5}{2}{x}^{2}+4x+1. If a_{1} and a_{5} are exactly two extreme points of f(x), find the value of a_{3}.

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