
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 a quadratic function $f(x) = x^2 + mx + n$.
1. If $f(x)$ is an even function and its minimum value is 1, find the expression for $f(x)$.
2. Based on (1), for the function $g(x) = \frac{6x}{f(x)}$, solve the inequality $g(2^x) > 2^x$ for $x$.
3. Let $h(x) = |f(x)|$ and assume that for $x \in [-1, 1]$, the maximum value of $h(x)$ is $M$, such that $M \geq k$ holds true for any real numbers $m$ and $n$. Find the maximum value of $k$.

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