
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
The function  $g\left(x\right)$  is defined as  $\sqrt{\dfrac{x}{2}}$  for all positive  $x$ .  $ $   $$ g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right) $$    $ $ can be expressed as  $\cos(b)$  using degrees, where  $0^\circ < b < 90^\circ$  and  $b = p/q$  for some relatively prime positive integers  $p, q$ . Find  $p+q$ .

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