
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
1. \([x]\) is the integer part of the number \(x\) (the greatest integer that does not exceed \(x\)), and \(\{x\} = x - [x]\) is the fractional part of the number \(x\).

For which \(a\) does the system
\[
\begin{cases} 
2x - [x] = 4a + 1 \\ 
4[x] - 3\{x\} = 5a + 15 
\end{cases}
\]
have a solution? Find such \(x\).

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