
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 function \( f(x) = \frac{1}{4^x + 2} \) for \( x \in \mathbf{R} \), points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \) lie on the graph of the function \( f(x) \), and the midpoint \( P \) of the line segment \( P_1P_2 \) has an x-coordinate of \( \frac{1}{2} \).

1. If the sequence \(\left\{a_n\right\}\) has a general term \( a_n = f \left( \frac{n}{m} \right) \) for \( m \in \mathbf{N} \) and \( n = 1, 2, \ldots, m \), find the sum \( S_m \) of the first \( m \) terms of the sequence \(\left\{a_n\right\}\).
2. If \( m \in \mathbf{N} \) and the inequality \(\frac{a^m}{S_m} < \frac{a^{m+1}}{S_{m+1}} \) always holds, determine the range of the real number \( a \).

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