
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 the function $f(x)$ satisfies $f(x) + f(-x) = 0$ and $f(x - 1) = f(x + 1)$ for all real numbers $x \in \mathbb{R}$. If $f(x) = a^x + b$ for $x \in [0, 1)$, where $a > 0$ and $a \neq 1$, and $f(\frac{3}{2}) = 1 - \sqrt{2}$.

1. Find the expression for $f(x)$ when $x \in [-1, 1]$.
2. Determine the number of real solutions to the equation $f(x) - |\log_4 x| = 0$.

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