
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
A function  $S(m, n)$  satisfies the initial conditions  $S(1, n) = n$ ,  $S(m, 1) = 1$ , and the recurrence  $S(m, n) = S(m - 1, n)S(m, n - 1)$  for  $m\geq 2, n\geq 2$ . Find the largest integer  $k$  such that  $2^k$  divides  $S(7, 7)$ .

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