
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 \( a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6} \) be non-negative integers and satisfy
\[
\begin{cases}
a_{1} + 2a_{2} + 3a_{3} + 4a_{4} + 5a_{5} + 6a_{6} = 26 \\
a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} = 5
\end{cases}
\]
If \( c \) is the number of solutions to the system of equations, determine the value of \( c \).

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