
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
Let the function $f(x) = e^x - e^{-x}$, and $g(x) = \log(mx^2 - x + \frac{1}{4})$. If for any $x_1 \in (-\infty, 0]$, there exists $x_2 \in \mathbb{R}$ such that $f(x_1) = g(x_2)$, determine the minimum value of the real number $m$.

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