
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 the number of solutions in positive integers \(\left(k; a_{1}, a_{2}, \ldots, a_{k}; b_{1}, b_{2}, \ldots, b_{k}\right)\) to the equation
\[ 
a_{1}\left(b_{1}\right) + a_{2}\left(b_{1}+b_{2}\right) + \cdots + a_{k}\left(b_{1}+b_{2}+\cdots+b_{k}\right)= 7.
\]

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