
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
Solve the system of inequalities:

\[
\left\{
\begin{array}{l}
|x-2a| \leq 3 \\
\log_{2}(x+a) \leq 2 
\end{array}
\right.
\]

which is equivalent to:

\[
\left\{
\begin{array}{l}
-5 \leq x-2a \leq 5 \\
0 < x+a \leq 4 
\end{array}
\right.
\]

and further simplifies to:

\[
\left\{
\begin{array}{l}
2a-5 \leq x \leq 2a+5 \\
-a < x \leq 4-a 
\end{array}
\right.
\]

The system has a unique solution if \(2a - 5 = 4 - a\), which gives \(a = 3\). In this case, the only solution for \(x\) is \(x = 1\).

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