
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 function $f(x)$ defined on $R$ such that $f(-x) + f(x) = 0$, and the graph of $y = f(x+1)$ is symmetric about the $y$-axis. For $x \in [0,1]$ and any $k \in [0,1]$, $f(x)$ satisfies $f(kx) + 1 = [f(x) + 1]^k$, and $f(1) = -\frac{1}{2}$. Evaluate $f\left(-\frac{9}{2}\right)$.

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