
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 the four functions: \( f(x) = \ln \left( x + \sqrt{x^2 + 1} \right) \), \( f(x) = \frac{1 + \sin 2x + \cos 2x}{1 + \sin 2x - \cos 2x} \), \( f(x) = \frac{ax}{x^2 - 1} \) (\( a \in \mathbb{R} \)), and \( f(x) = \begin{cases} 1 - 2^{-x}, & \text{if } x \geq 0 \\ 2^x - 1, & \text{if } x < 0 \end{cases} \), determine the number of functions that are odd.

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