
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 conditions:
\[
\left\{
\begin{array}{l}
a < 0 \\
b < 0
\end{array}
\right.
\Rightarrow
\left\{
\begin{array}{l}
49^{x+1} - 50 \cdot 7^x + 1 < 0 \\
\log_{x+\frac{5}{2}}\left|x+\frac{1}{2}\right| < 0
\end{array}
\right.
\]

rewrite the problem as:
\[
\left\{
\begin{array}{l}
\left( 7^x - 1 \right) \left( 7^x - 7^{-2} \right) < 0 \\
\left( x + \frac{5}{2} - 1 \right) \left( \left| x + \frac{1}{2} \right| - 1 \right) < 0 \\
x + \frac{5}{2} > 0, x + \frac{5}{2} \neq 1 \\
x + \frac{1}{2} \neq 0
\end{array}
\right.
\]

The final system after transformations is:
\[
\left\{
\begin{array}{l}
(7 - 1)x (7 - 1)(x + 2) < 0 \\
\left( x + \frac{3}{2} \right) \left( x - \frac{1}{2} \right) \left( x + \frac{3}{2} \right) < 0 \\
x + \frac{5}{2} > 0, x \neq -\frac{3}{2}, x \neq -\frac{1}{2}
\end{array}
\right.
\]

Combine the results of solving these systems to obtain the final answer.

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