
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 $a\neq 0$, let $f(x)=2\sqrt{3}\sin x \cos x + 2\cos^2 x - 1 - a$ and $g(x)=a\log_2(x+3)-2$. If there exists $x_1 \in [0,\frac{\pi}{2}]$ such that for all $x_2 \in [1,5]$, $f(x_1)=g(x_2)$, then the range of real number $a$ is ______.

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