
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 functions $f(x)= \begin{cases} 1, &x\text{ is rational} \\ 0, &x\text{ is irrational} \end{cases}$ and $g(x)= \begin{cases} 1, &x\text{ is irrational} \\ 0, &x\text{ is rational} \end{cases}$, find $f(g(x))=$ _______ and $g(f(x))=$ _______ when $x \in \mathbb{R}$.

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