
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  $a_1=2021$  and for  $n \ge 1$  let  $a_{n+1}=\sqrt{4+a_n}$ . Then  $a_5$  can be written as   $$ \sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}}, $$   where  $m$  and  $n$  are positive integers. Find  $10m+n$ .

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