
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 $a$ and $b \in \mathbb{R}$, the line $y = ax + b + \frac{\pi}{2}$ is tangent to the graph of the function $f(x) = \tan x$ at $x = -\frac{\pi}{4}$. Let $g(x) = e^x + bx^2 + a$. If the inequality $m \leq g(x) \leq m^2 - 2$ holds true for all $x$ in the interval $[1, 2]$, find the minimum value of the real number $m$.

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