
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
Find all real values of the parameter  $a$  for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these values of  $a$ .

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