
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 $f(x)=x^2+2x$ and $g(x)=(\frac{1}{2})^x+m$, for any $x_1 \in [1, 2]$, there exists an $x_2 \in [-1, 1]$ such that $f(x_1) \ge g(x_2)$. Determine the range of values for the real number $m$.

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