
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
Find all real  $a$  such that the equation  $3^{\cos (2x)+1}-(a-5)3^{\cos^2(2x)}=7$  has a real root.

<details><summary>Remark</summary>This was the statement given at the contest, but there was actually a typo and the intended equation was  $3^{\cos (2x)+1}-(a-5)3^{\cos^2(x)}=7$ , which is much easier.</details>

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