
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 $f(x)$ is an even function and $g(x)$ is an odd function on $\mathbb{R}$, and $f(x) + g(x) = e^x$, find the range of values for the real number $m$ such that for any $x_1 \in [0, 1]$ there exists $x_2 \in [0, 1]$ such that $f(x_1) = m g(x_2)$.

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