
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
Given a sequence $\{a_n\}$ satisfying $a_1=a$, $a_{n+1}=(2|\sin \frac {n\pi}{2}|-1)a_n+2n$.  
(I) Please write down the values of $a_2$, $a_3$, $a_4$, $a_5$;  
(II) Conjecture the general formula of the sequence $\{a_n\}$, no proof required;  
(III) Using the conjectured result from (II), calculate the sum of the first $120$ terms of the sequence $\{a_n\}$.

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