
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 differentiable function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+2)-f(x)=2f(1)$, and the graph of $y=f(x+1)$ is symmetric about the line $x=-1$. Also, when $x \in [2,4]$, $f(x)=x^2+2xf'(2)$, determine the relationship between $f(-\frac{1}{2})$ and $f(\frac{16}{3})$.

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