
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 the function $f(x) \in \{\sin x, |\log_2 x|, \log_2 |x|, x^{\frac{1}{2}}\}$ is an even function.
1) Write out the analytical expression of the function $f(x)$ that satisfies the condition (no need to explain the reason);
2) Let $g(x) = m \cdot 2^{f(x)} + x^2 (m \in \mathbb{R})$;
   a) Find the range of values of the real number $m$ if the function $g(x)$ is a decreasing function on the interval $(-\infty, -2)$;
   b) Determine whether $g(x) > \frac{x}{4} + \frac{1}{x}$ holds true on $x \in [1, 2]$ when $m > \frac{1}{4}$, and explain the reason.

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