
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 four functions $f_1(x)=x$, $f_2(x)=\sin x$, $f_3(x)=\cos x$, and $f_4(x)=\lg (x+ \sqrt {1+x^{2}})$. If we take any two of these functions and multiply them together to form a new function, what is the probability that the new function is an odd function?

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