
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
Consider a string of $n$ $7$s, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. How many values of $n$ are possible if the inserted $+$ signs create a sum of $7350$ using groups of $7$s, $77$s, $777$s, and possibly $7777$s?

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