
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
Let \( a, b \), and \( c \) be real numbers. Consider the system of simultaneous equations in variables \( x \) and \( y \):
\[ 
\begin{aligned}
a x + b y &= c - 1 \\
(a+5) x + (b+3) y &= c + 1 
\end{aligned}
\]

Determine the value(s) of \( c \) in terms of \( a \) such that the system always has a solution for any \( a \) and \( b \).

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